








































Class_ T 5 55 

Book . IB )'! 5. 
Copyright iV’_ 


COPYRIGHT DEPO&ET. 










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ENGINEERING 
DESCRIPTIVE GEOMETRY 




Engineering 
Descriptive Geometry 


A Treatise on Descriptive Geometry as the Basis of Mechanical 
Drawing, Explaining Geometrically the Operations 
Customary in the Draughting Room 


by 

F. W. BARTLETT 

COMMANDER, U. S. NAVY 

HEAD OF DEPARTMENT OF MARINE ENGINEERING AND NAVAL CONSTRUCTION 
AT THE UNITED STATES NAVAL ACADEMY 

AND 

THEODORE W. JOHNSON 

A.B., M.E. 

PROFESSOR OF MECHANICAL DRAWING, UNITED STATES NAVAL ACADEMY 
MEMBER OF AMERICAN SOCIETY OF MECHANICAL ENGINEERS 



ANNAPOLIS, MD., U. S. A. 

1910 



Copyright, 1910, by 
T. W. Johnson 


\ 


• * 

* c 

• • * 


JSorb Q0afftmore (prece 

BALTIMORE, MD., tJ. S. A. 


C Cl. A 2 6 I 0 5 2 


PEEFACE. 


The aim of this work is to make Descriptive Geometry an 
integral part of a course in Mechanical or Engineering Drawing. 

The older books on Descriptive Geometry are geometrical rather 
than descriptive. Their authors were interested in the subject as a 
branch of mathematics, not as a branch of drawing. 

Technical schools should aim to produce engineers rather than 
mathematicians, and the subject is here presented with the idea 
that it may tit naturally in a general course in Mechanical Drawing. 
It should follow that portion of Mechanical Drawing called Line 
Drawing , whose aim is to teach the handling of the drawing instru¬ 
ments, and should precede courses specializing in the various 
branches of Drawing, such as Mechanical, Structural, Architectural, 
and Topographical Drawing, or the “ Laying Off ” of ship lines. 

The various branches of drawing used in the different industries 
may be regarded as dialects of a common language. A drawing is 
but a written page conveying by the use of lines a mass of informa¬ 
tion about the geometrical shapes of objects impossible to describe 
in words without tedium and ambiguity. In a broad sense all these 
branches come under the general term Descriptive Geometry. It 
is more usual, however, to speak of them as branches of Engineer¬ 
ing Drawing, and that term may well be used as the broad label. 

The term Descriptive Geometry will be restricted, therefore, to 
the common geometrical basis or ground work on which the various 
industrial branches rest. This ground work of mathematical laws 
is unchanging and permanent. 

The branches of Engineering Drawing have each their own 
abbreviations, and special methods adapting them to their own 
particular fields, and these conventional methods change from time 
to time, keeping pace with changing industrial methods. 

Descriptive Geometry, though unchanged in its principles, has 
recently undergone a complete change in point of view. In 
changing its purpose from a mathematical one to a descriptive one, 


VI 


Preface 


from being a training for the geometrical powers of a mathematician 
to being a foundation on which to build up a knowledge of some 
branch of Engineering Drawing, the number and position of the 
planes of projection commonly used are altered. The object is now 
placed behind the planes of projection instead of in front of them, 
a change often spoken of as a change from the “ 1st quadrant ” to 
the “ 3d quadrant,” or from the French to the American method. 
We make this change, regarding the 3d quadrant method as the 
only natural method for American engineers. All the principles of 
Descriptive Geometry are as true for one method as for the other, 
and the industrial branches, as Mechanical Drawing, Structural 
Drawing, etc., as practiced in this country, all demand this method. 

In addition, the older geometries made practically no use of a 
third plane of projection, and we take in this book the further step 
of regarding the use of three planes of projection as the rule, not 
the exception. To meet the common practice in industrial branches, 
we use as our most prominent method of treatment, or tool, the use 
of an auxiliary plane of projection, a device which is almost the 
draftsman’s pet method, and which in books is very little noticed. 

As the work is intended for students who are but just taking up 
geometry of three dimensions, in order to inculcate by degrees a 
power of visualizing in space, we begin the subject, not with the 
mathematical point in space but with a solid tangible object shown 
by a perspective drawing. ATo exact construction is based on the 
perspective drawings which are freely used to make a realistic ap¬ 
pearance. As soon as the student has grasped the idea of what 
orthographic projection is, knowledge of how to make the projection 
is taught by the constructive process, beginning with the point and 
passing through the line to the plane. To make the subject as 
tangible as possible, the finite straight line and the finite portion of 
a plane take precedence over the infinite line and plane. These 
latter require higher powers of space imagination, and are therefore 
postponed until the student has had time to acquire such powers 
from the more naturally understood branches of the subject. 

F. W. B. 

T. W. J. 


March, 1910. 


CONTENTS. 

CHAPTER PAGE 

I. Nature of Orthographic Projection. 1 

II. Orthographic Projection of the Finite Straight Line. .. 18 

III. The True Length of a Line in Space. 27 

IV. Plane Surfaces and Their Intersections and Develop¬ 

ments . 38 

V. Curved Lines . 49 

VI. Curved Surfaces and Their Elements. 62 

VII. Intersections of Curved Surfaces. 70 

VIII. Intersections of Curved Surfaces; Continued. 81 

IX. Development of Curved Surfaces. 91 

X. Straight Lines of Unlimited Length and Their Traces. . 98 

XI. Planes of Unlimited Extent: Their Traces. 108 

XII. Various Applications . 121 

XIII. The Elements of Isometric Sketching. 133 

XIV. Isometric Drawing as an Exact System. 142 

Set of Descriptive Drawings. 149 























CHAPTER I. 

NATURE OF ORTHOGRAPHIC PROJECTION. 


1. Orthographic Projection. —The object of Mechanical Draw¬ 
ing is to represent solids with such mathematical accuracy and 
precision that from the drawing alone the object can be built or 
constructed without deviating in the slightest from the intended 
shape. As a consequence the “ working drawing ” is the ideal 
sought for, and any attempt at artistic or striking effects as in 
“ show drawings ” must be regarded purely as a side issue of minor 
importance. Indeed mechanical drawing does not even aim to 
give a picture of the object as it appears in nature, but the views 
are drawn for the mind, not the eye. 

The shapes used in machinery are bounded by surfaces of mathe¬ 
matical regularity, such as planes, cylinders, cones, and surfaces 
of revolution. Thev are not random surfaces like the surface of a 
lump of putty or other surfaces called “ shapeless.” These definite 
shapes must be represented on the flat surface of the paper in an 
unmistakable manner. 

The method chosen is that known as " orthographic pro jection” 
If a plane is imagined to be situated in front of an object, and 
from any salient point, an edge or corner, a perpendicular line, 
called a “ projector,” is drawn to the plane, this line is said to 
project the given point upon the plane, and the foot of this perpen¬ 
dicular line is called the projection of the given point. If all 
salient points are projected by this method, the orthographic draw¬ 
ing of the object is formed. 

2. Perspective Drawing.—The views we are accustomed to in 
artistic and photographic representations are “ Perspective Views.” 
They seek to represent objects exactly as they appear in nature. 
In their case a plane is supposed to be erected between the human 
eye and the object, and the image is formed on the plane by sup¬ 
posing straight lines drawn from the eye to all salient points of 


2 


Engineering Descriptive Geometry 


the object. Where these lines from the eye, or “ Visual Rays,” as 
they are called, pierce the plane, the image is formed. 

Fig. 1 represents the two contrasted methods applied to a simple 
object, and the customary nomenclature. 

An orthographic view is sometimes called an “ Infinite Perspec¬ 
tive View,” as it is the view which could only be seen b}^ an eye at 
an infinite distance from the object. “ The Projectors ” may then 
be considered as parallel visual rays which meet at infinity, where 
the eye of the observer is imagined to be. 

Projector 

A 

K- > 
fej UJ 

/P o 

Projector* 

: View. 

3. The Regular Orthographic Views. —Since solids have three 
“ dimensions,” length, breadth and thickness, and the plane of the 
paper on which the drawing is made has but two, a single ortho¬ 
graphic view can express two only of the three dimensions of the 
object, but must always leave one indefinite. Points and lines at 
different distances from the eye are drawn as if lying in the same 
plane. Prom one view only the mind can imagine them at dif¬ 
ferent distances by a kind of guess-work. If two views are made 
from different positions, each view may supplement the other in 
the features in which it is lacking, and so render the representa¬ 
tion entirely exact. Theoretically two views are always required 
to represent a solid accurately. 

To make a drawing all the more clear, other views are generally 
advisable, and three views may be taken as the average requirement 
for single pieces of machinery. Six regular views are possible, 
however, and an endless number of auxiliary views and “ sections ” 
in addition. For the present, we shall consider only the “ regular 
views,” which are six in number. 




Orthographic 


Fig. 1. 














Nature of Orthographic Projection 3 

4. Planes of Projection.—A solid object to be represented is 
supposed to be surrounded by planes at short distances from it, the 
planes being perpendicular to each other. From each point of 
every salient edge of the object, lines are supposed to be drawn 
perpendicular to each of the surrounding planes, and the succes¬ 
sion of points where these imaginary projecting lines cut the planes 
are supposed to form the lines of the drawings on these planes. 
One of the planes is chosen for the plane of the paper of the actual 
drawing. To bring the others into coincidence with it, so as to 
have all of them on one flat s*heet, they are imagined to be unfolded 
from about the object by revolving them about their lines of inter¬ 
section with each other. These lines of intersection, called “ axes 
of projection,” separate the flat drawing into different views or 
elevations. 
































































4 


Engineering Descriptive Geometry 



Fig. 2 is a true perspective drawing of a solid object and the 
planes as they are supposed to surround it. This figure is not a 
mechanical drawing, but represents the mental process by which 
the mechanical drawing is supposed to be formed by the projection 
of the views on the planes. In this case the planes are supposed 
to be in the form of a perfect cube. The top face of the cube shows 
the drawing on that face projected from the solid by fine dotted 
lines. Eemember that these fine dotted lines are supposed to be 
perpendicular to the top plane. This drawing on the top plane is 
called the “ plan.” On the front of the cube the “ front view ” or 
“ front elevation ” is drawn, and on the right side of the cube is 

















Nature of Orthographic Projection 


o 


the “ right side elevation.” Three other views are supposed to be 
drawn on the other faces of the cube, but they are shown on Fig. 
2a, which is the perspective view of the cube from the opposite 
point of view, that is, from the back and from below instead of 
from in front and from above. 

This method of putting the object to be drawn in the center of 
a cube of transparent planes of projection is a device for the im¬ 
agination only. It explains the nature of the “ projections/’ or 
“ views,” which are used in engineering drawing. 

5. Development or Flattening Out of the Planes of Projection.— 
Now imagine the six sides of the cube to be flattened out into one 
plane forming a grouping of six squares as in Fig. 3. What we 




Plan 



S’ 


X 

V 


0 


s 



V' 



N 

1 N- 

s > 

_LLL 








4 



i i 

i 

-i 

i 

_i_i_ 


Left Side 

E LEVATI ON 

Front 

Elevation 

Z 

(Right) Side 
Elevation 

Back 

Elevation 




Bottom 

View 


Fig. 3. 


have now is a description or mechanical drawing of the object 
showing six “ views.” The object itself is now dispensed with and 
its projections are used to represent it. These six views are what 
we call the “regular views.” With one slight change they cor¬ 
respond to the regular set of drawings of a house which architects 
make. 

2 


y 































6 


Engineering Descriptive Geometry 


The set of six “ regular ” projections would not be altered by 
passing the transparent planes at unequal distances from each 
other^ so long as they surround the object and are mutually per¬ 
pendicular. They may form a rectangular parallelopiped instead 
of a cube without altering the nature of the views. 

It will be noticed also that views on opposite faces of the cube 
differ but little. Corresponding lines in the interior may in one 
case be full lines and in the other “ broken lines.” Broken lines 
(formed by dashes about -J" long, with spaces of yV") represent 
parts concealed by nearer portions of the object itself. All edges 
project upon the plane faces of the cube, forming lines on the draw¬ 




ings, the edges concealed by nearer portions of the object forming 
broken lines. 

6. The Reference Planes and Principal Views.—In drawings of 
parts of machinery six regular views are usually unnecessary. The 
three views shown in Fig. 2 are the “ Principal Views,” and others 
are needed only occasionally. The planes of those views are the 
“ Reference Planes/’ 

These views, when flattened from their supposed position about 
the object into one plane, give the grouping in Fig. 4. 

Another arrangement of the same views, obtained by unfolding 
the planes of the cube in a different order, is shown in Fig. 5. 
These two arrangements are standard in mechanical drawing, and 
are those most used. 



































Nature op Orthographic Projection 


7 


7. The Nomenclature.—The nomenclature adopted is as follows: 
The “ Reference Planes,” or three principal planes of projection, 
are called from their position, the Horizontal Plane, or H, the 
Vertical Plane, or Y, and the (right) Side Plane, or §. The plane 
S is by some called the “ Profile Plane.” The point 0 (Fig. 2), 
in which they meet, is the “Origin” of coordinates. The line 
OX, in which and V intersect, is called the “ Axis of X,” or 
“ Ground Line." The line OY in which ff and § meet, is called 
the “ Axis of Y; ' and the line OZ, in which V and § meet, is 
called the “ Axis of ZY The three axes together are called the 
“ Axes of Projection.” 

Since drawings are considered as held vertically before the face, 
it is considered that the plane V coincides at all times with the 
“ Plane of the Paper.' 7 In unfolding the planes from their posi¬ 
tions in Fig. 2 to that in Fig. 4, it is considered that the plane H 
has been revolved about the axis of X (line OX), through an angle 
of 90°, until it stands vertically above V- In the same way § is 
considered to be revolved about the line OZ, or axis of Z, until it 
takes its place to the right of V- 

The arrangement in Fig. 5 corresponds to a different manner of 
revolving the plane §. It is revolved about the axis of Y (OY) 
until it coincides with the plane ff, and is then revolved with fj, 
about the axis of X, until both together come into the plane of the 
paper, or V- 

The three other faces of the original cube of planes of projection 
are appropriately called fj', and S'. • On account of the simi¬ 
larity of the views on them, to those on J-J, V and §, they are but 
little used. S r alone is fairly common since a grouping of planes 
H, V and S' is at times more convenient than the standard group 
M, V and S- 

8. Meaning of “ Descriptive Geometry.”—The aim of Engineer¬ 
ing or Mechanical Drawing is to represent the shapes of solid 
objects which form parts of structures or machines. It shows 
rather the shapes of the surfaces of the objects, surfaces which are 
usually composed of plane, cylindrical, conical, and other surfaces. 
In the drawing room, by the application of mathematical laws and 
principles, views are constructed. These are usually Plan, Front 


8 


Engineering Descriptive Geometry 


Elevation, and Side Elevation, and are exactly such views as would 
be obtained if the object itself were put within a cage of trans¬ 
parent planes, and the true projections formed. 

It is these mathematical laws or rules which form the subject 
known as Descriptive Geometry. A drawing made in such a way 
as to bring out clearly these fundamental laws of projection, by the 
use of axes of projection, etc., may be conveniently called a “ De¬ 
scriptive Drawing.” 

In the practical application of drawing to industrial needs, 
short-cuts, abbreviations, and special devices are much used (their 
nature depending on the special branch of industr}^ for which the 
drawing is made). In addition, the axes of projection are usually 
omitted or left to the imagination, no particular effort being made 
to show the exact mathematical basis provided the drawing itself 
is correct. Such a drawing is a typical “ Mechanical Drawing.” 
By the addition of axes of projection and similar devices, it friay 
be converted into a strict “ Descriptive Drawing.” 

9. The Descriptive Drawing of a Point in Space.—The imagi¬ 
nary process of making a descriptive drawing consists in putting 
the object within a cube of transparent planes, and projecting 
points and lines to these planes. In practice the projections are 
formed all on a single sheet of paper, which is kept in a perfectly 
flat shape, by the application of rules of a geometrical kind de¬ 
rived from the imaginary process. The key to the practical pro¬ 
cess is in these rules. The first subject of exact investigation 
should be the manner of representing a point in space by its pro¬ 
jections and the fixing of its position as regards the “ reference 
planes ” by the use of coordinate distances. 

Figs. 6 and 7 show the imaginary and the practical processes of 
representing P by its projections. 

Fig. 6 is a perspective drawing showing the cube of planes, or 
i ather the three sides of the cube regularlv used for reference 
planes. The cube must he of such size that the point P falls well 
within it. The perpendicular projectors of P are PP h , PP V and 
PP» The origin and the axes of projection are all marked as on 
Fig. 2. 


Nature of Orthographic Projection 


9 


In Fig. 7 the “ field” of the drawings that part of the paper 
devoted to it, is prepared by drawing two straight lines at right 
angles to represent the axes of projection, lettering the horizontal 
line XOY s and the vertical one ZOYu. This field corresponds to 
that of Fig. 4, the outer edges of the squares being eliminated 
since there is no need to confine each plane to the size of any par¬ 
ticular cube. If more field is needed, the lines are simply ex¬ 
tended. It must be remembered that these axes are quite different 
from the coordinate axes used in plane analytical geometry, or 
graphic algebra. These divide the field of the drawing into four 



quadrants, of which three represent three different planes, mutu¬ 
ally perpendicular, the fourth being useful only for the purposes 
of construction. 

Usually the upper left quadrant, the “ North-West," represents 
m; the lower left quadrant, or “ South-West,” represents V, and 
the lower right quadrant, or u South-East,” represents S- 

On occasion the axes may be lettered XOZ s horizontally and 
Z v OY vertically, to correspond to Fig. 5, the upper right quadrant 
now representing §. 

10. Coordinates of a Point in Space.—A point in space is 
located by its perpendicular distances from the three planes of 
projection, that is to say, by the length of its projectors. These 






















10 


Engineering Descriptive Geometry 


distances are called the coordinates of the point, and are designated 
by x, y and z. In the example given, these values are 2, 3 and 1. 
In Fig. 6 PP S , the § projector of P, is two units long, or x—2. 
The perpendicular distance to the plane V; the V projector, PP V , 
is three units long, y— 3. In the same way PPh, the H projector, 
is one unit long, z — 1. 

In describing the point P, it is sufficient to state that it is the 
point for which x—2 , y= 3, and z — 1. This is abbreviated con¬ 
veniently by calling it the point P (2, 3,1), the coordinates, given 
in the bracket, being taken always in the order x, y, z. 

The projectors, the true coordinate distances, are shown in Fig. 
6 by lines of dots, not dashes. 

If in each plane H, V and perpendicular lines are drawn 
(dashes, not dots) from the projections of P to the axes, we shall 
have the lines Pne and Pj,f, P v e and P v g, P s g and P s f. These lines 
meet in pairs at e, g, and f, forming a complete rectangular paral- 
lelopiped of which P and 0 are the extremities of a diagonal. The 
other corners of the parallelopiped are Ph, P v , Ps, v, f and g. 

Each coordinate, x, y and z, appears in four places along four 
edges of the parallelopiped, as is marked in Fig. 6. 

The distances x, y and z are all considered positive in the case 
shown. 

In Fig. 7, the descriptive drawing of the point P, P itself does 
not appear, being represented by its projections, P h , P v and P a . 
The true projectors (shown in Fig. 6 by lines of dots) do not 
appear, but in place of each coordinate three distances equal to it 
do appear, so that in Fig. 7 x, y and 2 each appear in three places 
as is there marked. Thus x appears as P h fh, eO, and P v g. As all 
these are measured to the left from the vertical axis, ZOY h , it 
follows that Pnepv is a straight line, or P h is vertically above P v . 
It is often said that P v “projects vertically to P h . In the same 
way P v “ projects horizontally 99 to P s . The distance y appears as 
eP h , Ofh, Of s , and gP 8 . The point / appears double due to the 
axis of Y itself doubling. To represent the original coincidence of 
fn and f s , a quadrant of a circle with center at 0 is often used to 
connect them. 


Nature of Orthctgraphic Projection 


11 


11. Three Laws of Projection for H, V anc * S-— The three rela¬ 
tions shown by Fig. 7 amount to three laws governing the pro¬ 
jections of a point in the three views, and must always be rigidly 
observed. They may seem easy and obvious when applied to one 
point, but when dealing w T ith a multitude of points it is not easy 
to observe them unfailingly. 

They may be thus tabulated: 

(1) Pn must be vertically above P v . 

(2) P s must be on the same horizontal line as P v . 

(3) P s must be as far to the right of OZ as Pn is above OX. 

From these laws it follows that if two projections of a point are 

given, the third is easily found. In Fig. 7, if two of the corners 
of the figure PnfnfsPsPv are given, the figure can be graphically 
completed. Much of the work of actual mechanical drawing con¬ 
sists in correctly locating two of the projections of a point by plot¬ 
ting or measuring, and of finding the other projection by the appli¬ 
cation of these laws or of this construction. Constant checking 
of the points between the various views of a drawing is a vital prin¬ 
ciple in drawing. 

On the drawing board the horizontal projection of P v to P s is 
naturally done by the T-square alone, and the vertical projection 
of Pn to P v by T-square and triangle. There are two methods of 
carrying out the third law in addition to the graphical construc¬ 
tion of Fig. 7. Fig. 8 shows a graphical method which makes use 
of a 45° line, OL, in the construction space, instead of the quad¬ 
rant of a circle. It is easier to execute, but the meaning is not so 
clearly shown. The third method is by the use of the dividers 
directly to transfer the x coordinate from whichever place it is 
first plotted, to the other view in which it appears. 

12. Paper Box Diagrams.—When studying a descriptive draw¬ 
ing, such as Fig. 8, imagine as you look at P v that the real point P 
lies back of the paper, at a distance equal to ePj,. 

Whenever figures in the text following seem hard to grasp, carry 
out the following scheme. Trace the figure on thin paper, or on 
tracing cloth. Using Fig. 8 as an example, and supposing it to 
have been traced on semitransparent paper, hold the paper before 
you and fold the top half back 90° on the line XOY s . Then, view- 


12 


Engineering Descriptive Geometry 


ing P h from above, imagine the true point P to lie below the paper 
at a distance equal to eP v , in the same way as you imagine P to 
lie back of P v at a distance equal to eP h . 

After flattening the paper, fold the right half back 90° on the 
line ZOYh, and, viewing P s from the right, imagine P to lie back 
of P s a distance P v g. Finally, crease the paper on the line OL, 
OL itself forming a groove, not a ridge, and bend the paper on all 



the creases at once, so that ff and S fold back into positions at 
right angles to V an( f to each other at the same time. 

The “ construction space ” Y h OY s is thus folded away inside 
and OYn and OY s come in contact with each other. Fig. 9 shows 
the final folding partly completed. 

No diagram, however complicated, can remain obscure if studied 
from all sides in this manner. 

To have a convenient name, these space diagrams may be called 
“ Paper Box Diagrams/’ 










Nature of Orthographic Projection 


13 


Figs. 4 and 5 make good paper box diagrams, while Fig. 3 may 
be traced and folded into a perfect cube which, if held in proper 
position, will give the exact views shown in Figs. 2 and 2a, omit¬ 
ting the solid object supposed to be seen in the center of those 
figures. 

13. Zero Coordinates.—Points having zero coordinates are some¬ 
times perplexing. If one coordinate is zero, the point in question 
is on one of the reference planes, and indeed coincides with one of 
its own projections. Since x is the length of the orthographic 
projector of the point P upon the plane §, if x=Q, this projector 
disappears and the point P and its § projection P s coincide. If 



the point Q (0, 3, 1) is to be plotted it will be found to coincide 
with P s in Fig. 6. The descriptive drawing will correspond with 
Fig. 7 with all lines to the left of ZOYn omitted, and with the let¬ 
tering changed as follows: For P s put Q s (and Q), for f h put Qn, 
for g put Q v . The student should make this diagram on cross- 
section paper and should study out for himself the similar cases for 
the points Q' (2, 0,1) [P v in Fig. 6] and Q" (2, 3, 0) [P h in Fig. 
6] and should proceed from them to more general cases, assuming 
ordinates at will, using cross-section paper for rapid sketch work 
of this kind. 

If two coordinates are zero, the point lies on one of the axes, 
on that axis, in fact, which corresponds to the ordinate which is not 
zero. Thus the point R (2, 0, 0) is the point e of Fig. 6, Rj, and 
R v are at e, and R s is at 0. 









Pig. 10. 















































































































































































14444 



Fig. 10a. 





































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































16 


Engineering Descriptive Geometry 


Wire-mesh Cage. 

If possible, it is very desirable to have cages similar to Fig. 10, 
formed of wire-mesh screens, representing the planes H, V* S an d 
S'. On these screens chalk marks may be made and the planes, 
being hinged together, may afterward be brought into coincidence 
with V, as represented in Fig. 10 a. 

In order to plot points in space within the cage, pieces of wire 
about 20 inches long, with heads formed in the shape of small 
loops or eyes, are used as point markers. They may be set in holes 
drilled in the base of the cage at even spaces of 1" in each direc¬ 
tion, so that a marker may be set to represent any point whose x 
or y coordinates are even inches. To adjust the marker to a re¬ 
quired z coordinate, it may be pulled down so that the wire projects 
through the base, lowering the head the required amount, z may 
vary fractionally. 

In Fig. 10 a point marker is set to the point P (11, 4, 6), and 
the lines on the screens have been put on with chalk, to represent 
all the lines analogous to those of Fig. 6. 

Fig. 10a represents the descriptive drawing produced by the 
development of the screens in Fig. 10. It is analogous to Fig. 7. 

Several points may be thus marked in space and soft lead wire 
threaded through the loops, so that any plane figure may be shown 
in space, and its corresponding orthographic projections may be 
drawn on the planes in chalk. 

Problems I. 

1. Plot by the use of the wire markers the three points. A, B 
and C, whose coordinates are (5,12,11), (3, 3, 3), and (12, 4, 8), 
and draw the projections on the screens in chalk. By joining point 
to point we have a triangle and its projections. Use lead wire for 
joining the points, and chalk lines for joining the projections. 

2. Form the triangle as above with the following coordinates: 

(11,3,2), (12,6,12) and (14,12,7). 

3. Form the triangle as above with the following coordinates: 

(7,0,11), (9,9,0) and (2,2,3). 

4. Form the triangle as above with the following coordinates: 

(0,11,13), (14,3,3) and (14,13,0). 


Nature of Orthographic Projection 


17 


(The following examples may be solved on coordinate paper, or 
plotted in inches on the blackboard.) 

5. Make the descriptive drawing of a triangle in three views by 
plotting the vertices and joining them by straight lines. The 
vertices are the points A (1,10, 8), B (5, 6, 8), C (9, 2, 4). 

6. Make the descriptive drawing as above using the points 

A (12,2,5), B (0,8,6), C (4,6,0). 

7. Make the descriptive drawing as above using the points 

A (3,4,2), B (13,8,10), C (5,10,14). 

8. The four points A (3, 3, 3), B (3, 3,15), C (15, 3,15), and 
D (15, 3, 3) form a square. Make the descriptive drawing. Why 
are two projections straight lines only? What are the coordinates 
of the center of the square? 

9. The four points A (12,2,12), B (2,2,12), C (7,14,12), 
and D (7, 6, 2) are the corners of a solid tetrahedron. Make the 
descriptive drawing, being careful to mark concealed edges by 
broken lines. 

10. Make the descriptive drawing of the tetrahedron A (2, 3, 2), 
B (9,8,3), C (4,8,9), D (1,3,6), marking concealed edges by 
broken lines. 

11. Make the descriptive drawing of the tetrahedron A (3, 2, 4), 
B (6,8,2), (7 (8,1,8), 2? (2,7,8). 

12. Plot the points A (12, 7, 7), B (8,13, 5), C (2, 9. 2), and 
D (6,3,4). Why is the V projection a straight line? 

13. Make the descriptive drawing of the tetrahedron A (13, 5, 3), 
B (1, 5, 3), O (7, 2, 6), D (7, 8, 6). To which axis is the line AB 
parallel? To which axis is CD parallel? 

14. Plot and join the points A (11, 3, 3), B (3, 3, 3), C (7, 9, 7), 
and D (15, 9, 7). Do AC and BD meet at a point or do they pass 
without meeting? 


CHAPTER II. 


ORTHOGRAPHIC PROJECTION OF THE FINITE STRAIGHT 

LINE. 

14. The Finite Straight Line in Space: One not Parallel to any 
Reference Plane, or an “ Oblique Line.” —A line of any kind con¬ 
sists merely of a succession of points. Its orthographic projection 
is the line formed by the projections of these points. 

In the case of a straight line, the orthographic projection is 
itself a straight line, though in some cases this straight line may 
degenerate to a single point, as mathematicians express it. 



To find the H? Y and g projections of a finite straight line in 
space, the natural course is to project the extremities of the line 
on each reference plane and to connect the projections of the ex¬ 
tremities by straight lines. We shall not consider this as requir¬ 
ing proof here. It is common knowledge that a straight line cannot 
be held in any position that will make it appear curved, and ortho¬ 
graphic projection is, as shown by Fig. 1, only a special case of 
perspective projection. The strict mathematical proof is not ex¬ 
actly a part of this subject. 



































Orthographic Projection of Finite Straight Line 19 


The projectors from the different points of a straight line form 
a plane perpendicular to the plane of projection. This “ projector- 
plane,” of course, contains the given line. If the straight line is 
a limited or finite line the projector-plane is in the form of a 
quadrilateral having two right angles. Thus in Fig. 11 the H 
projectors of the straight line AB form the figure AA h BhB, having 
right angles at A h and B h . These projector-planes AA h B h B, 
AA S B S B, and AA V B V B are shown clearly in this perspective draw¬ 
ing, in which they are shaded. 

Fig. 12 is the descriptive drawing of the same line AB which has 
been selected as a “ line in space,” that is, as one which does not 
obey any special condition. In such general cases the projections 
are all shorter than the line itself. As drawn, the extremities are 



15. Line Parallel to One Reference Plane, or Inclined Line.— 

A line which is parallel to one reference plane, hut is not parallel 
to an axis, appears projected at its true length on that reference 
plane only. 

Figs. 13 and 14 show a line five units long, connecting the points 
A (1,2,2) and B (5,5,2). A h Bh is also five units in length but 
A V B V is but four and A S B S is three. The projector-plane AA h B h B 
is a rectangle. 

The student should construct on coordinate paper the two simi¬ 
lar cases. For example : the line 0 (4, 2,1 ), D (1, 2, 5) is parallel 
to V; E (2,1, 2), F (2, 5, 5) is parallel to §. 



























20 


Engineering Descriptive Geometry 


16. Line Parallel to One of the Axes and thus Parallel to 
Two Reference Planes.—If a finite straight line is parallel to one 
of the axes of projection, its projection on the two reference planes 
which intersect at that axis, will be equal in length to the line 
itself. Its projection on the other reference plane will be a simple 
point. 

Fig. 15 is the perspective drawing and Fig. 16 the descriptive 
drawing, of a line parallel to the axis of X, four units in length. 
Its extremities are the points A (1,2,2) and B (5,2,2). In H 



and V its projections are four units long. The projector-planes 
AAnBnB and AA V B V B are rectangles. The S projector-plane de¬ 
generates to a single line BAA S . It will he seen that the coordi¬ 
nates of the extreme points of the line differ only in the value of 
the x coordinate. In fact, any point on the line will have the 
y and z coordinates unchanged. It is the line ( x variable, 2, 2). 

The student should construct for himself descriptive drawings 
of lines parallel to the axis of Y and the axis of F, using prefer¬ 
ably “ coordinate paper ” for ease of execution. Good examples 
are the lines C (1,1,1), D (1, 5,1) and E (3,1, 1), F (3,1, 4). 
Points on the line CD differ only as regards the y coordinate. It 
is a line parallel to the axis of Y. EF is parallel to the axis of Z 
and 2 alone varies for different points along the line. 




































Orthographic Projection of Finite Straight Line 21 


17. Foreshortening’.—The projection of a line oblique to the 
plane of projection is shorter than the original line. This is 
called foreshortening. The H, V and § projections of Fig. 12, 
and the V and § projections of Fig. 14, are foreshortened. It is 
a loose method of expression, but a common one, to say that a line 
is foreshortened when it is meant that a certain projection of a 
line is shorter than the line itself. When subscripts are omitted 
and AiiBn is called AB, it is natural to speak of the line AB as 
appearing “ foreshortened ” in the plan view or projection on J-J. 
This inexact method of expression is so customary that it can 
hardly be avoided, but with this explanation no misconception 
should be possible. 

18. Inclined and Oblique Lines.—The words Inclined and Obli¬ 
que are taken generally to mean the same thing, but in this subject 
it becomes necessary to draw a distinction, in order to be able to 
specify without chance of misunderstanding the exact nature of a 
given line or plane. 

A line will be described as: 

Parallel to an axis, when parallel to any axis. As a special case a 
line parallel to the axis of ^ may be called simply vertical. 
Inclined, when parallel to a reference plane, but not parallel to an 
axis. The line AB, Fig. 13, is an illustration. 

Oblique, when not parallel to any reference plane or axis. The 
typical “ line in space ” is oblique. AB, of Fig. 11, illustrates 
this case. 

19. Inclined and Oblique Planes.—A plane will be called: 
Horizontal, when parallel to JfJ. The V projector-plane in Fig. 15 

is of this kind. 

Vertical, when parallel to V or §. The H projector-plane in Fig. 
15 is of this kind. 

Inclined, when perpendicular to one reference plane only. The H 
projector-plane of Fig. 13 is of this kind. 

Oblique, when not perpendicular to any reference plane. Planes 
of this kind will appear later on. 

The surface of the solid of Fig. 2 is composed of vertical, hori¬ 
zontal, and inclined planes (but no oblique plane). Its edges are 


22 


Engineering Descriptive Geometry 


lines, parallel to the axes of X, Y and Z; and inclined lines (be¬ 
cause parallel to §) ; but no oblique lines. 

20. The Point on a Given Line.—It is self-evident that if a 
given point is on a given line, all the projections of the point must 
lie on the projections of the line. 

If the middle point of a line AB is projected, as C in Fig. 17, its 
projections Cn, C v , and C s bisect the projections of the line. The 
reason for this appears when we consider the true shape of the 



projector-planes, all three of which appear distorted in the per¬ 
spective drawing, Eig. 17, and which do not appear at all on the 
descriptive drawing, Pig. 18. In Fig. 17 AA h BnB is a quadri¬ 
lateral, having right angles at An and Bn, it is therefore a trape¬ 
zoid. CCn is parallel to A An, and BBn, and since it bisects AB at C , 
it must also bisect A h Bn at Cn. The result of this is that in Fig. 
18, where the projections which do appear are of their true size, 
Cn bisects AhBh, C v bisects A V B V , and C s bisects A S B S . 

This principle applies to other points than the bisector. Since 
all Jfl projectors are parallel to each other, if any point divides AB 
into unequal parts, the projections of the point will divide the 
projectors of AB in parts having the same ratio. A point one- 


# 





















Orthographic Projection of Finite Straight Line 23 


tenth of the distance from A to B will, by its projections, mark 
off one-tenth of the distance on AnBu, A V B V , etc. 

The points illustrated in Figs. 17 and 18 are A (2,3,1), 
B (5, 5, 5) and C (3J, 4, 3). It will be noticed that the x coordi¬ 
nate of 0 is the mean of those of A and B, and the y and z coordi¬ 
nates of C also are the mean of the y and z coordinates of A and B. 

Unless all three of the projections of a point fall on the pro¬ 
jections of a line, the point is not in the given line. If one of the 
projections of the point be on the corresponding projection of the 
line, one other projection of both point and line should be ex¬ 
amined. If in this second projection it is found that the point 
does not lie on the line, it shows that the point in space lies in one 
of the projector-planes. 

Thus the point D in Fig. 18 has its V projection on A V B V , but 
its H and § projections are not on Aj,Bi, and A S B S . D is not a 
point in the line AB but is on the V projector-plane of AB, as is 
clearly shown on Fig. 17. 

In the case illustrated, I) v bisects B V C V . The plotting of the V 
projection of a point is governed only by its x and z coordinates. 
D v bisects B V C V because its x and 2 coordinates are the means of 
the x and 2 coordinates of B and C. The y coordinate of D, how¬ 
ever, has no connection with the y coordinates of B and C. 

21. The Isometric Diagram.—A device to obtain some of the 
realistic appearance of a true perspective drawing without the 
excessive labor of its construction is known as “ isometric ” draw¬ 
ing. 

A full explanation of this kind of drawing will follow later, 
but for present purposes we may regard it as a simplified per¬ 
spective of a cube in about the position of that in Figs. 2, 6, 11, 
etc., but turned a little more to the left. Vertical lines are un¬ 
changed. Lines which are parallel to the axis of X, and which in 
the perspective drawing incline up to the left at various angles, are 
all made parallel and incline at 30° to the horizontal. In the 
same way lines parallel to the axis of Y are drawn at 30° to the 
horizontal, inclining up to the right. 


24 


Engineering Descriptive Geometry 


i 

Fig. 19 shows the shape of a large cube divided into small unit 
cubes. In plotting points the same scale is used in all three direc¬ 
tions, that is, for distances parallel to all three axes. Fig. 19a 
shows the point P (2, 3,1) plotted in this manner, so that the 
figure is equivalent to the true perspective drawing, Fig. 6. 

It is not intended that the student should make any true per¬ 
spective drawing while studying or reciting from this book. If any 
of the space diagrams here shown by true perspective drawings 



must be reproduced, the corresponding isometric drawing should 
be substituted. 

For rapid sketch work, especially ruled paper, called “ isometric 
paper,” is very convenient. It has lines parallel to each of the 
three axes. With such paper it is easy to pick out lines correspond¬ 
ing to those of Fig. 19. 

An excellent exercise of this kind is to sketch on isometric 
paper the shaded solid shown in Fig. 2, taking the unit square of 
the paper for 1" and considering the solid to be cut from a 10" cube, 
the thickness of the Avails left being 2", and the height of the tri¬ 
angular portion being 6". The solid may be sketched in several 
positions. 















Orthographic Projection of Finite Straight Line 25 

Problems II. 

(For solution with wire-mesh cage, or cross-section paper, or on 

blackboard.) 

15. A line connects the points A (5,2,6) and B (5,12,6). 
What are the coordinates of the point C, the center of the line? 

What are the coordinates of a point D on the line, one-tenth of 
the way from A to B? 

16. Same with points A (6, 6, 2) and B (6, 6,12). 

17. Draw the line AB whose extremities are A (2, 7, 4) and 
B (14,2,4). On what view does its true length appear? What is 
this length? What are the coordinates of a point C on the line 
one-third of its length from A ? 

18. With the same line A (2,7,4), B (14,2,4), state what is 
the true shape of the fj projector-plane. Give length of each edge 
and state what angles are right angles. Same for V projector- 
plane. 

19. Same as Problem 18, with line A (4, 2, 2), B (4,11, 8). 

20. With the line of Problem 19, state what is the true shape of 
the ff and § projector-planes, giving length of each edge, and 
state which angles are right angles. 

21. Same as Problem 17, with line A (0, 4, 8), B (9, 4,1). 

22. The H projection of C (8, 2, 6) lies on the ff projection of 
the line A (10,1, 9), B (2, 5, 2). Is the point on the line? 

23. Same as Problem 22, with line A (2,1, 8), B (8,10, 5), and 
point C (4, 4, 7). 

24. A triangle is formed by joining the points A (6, 3,1), 
B (10,3,10) and C (2,3,10). In what view or views does the 
true length of AB appear? In what view or views does the true 
length of BC appear? Mark the center of the triangle (one-third 
the distance from the center of the base BC to the vertex A) and 
give its coordinates. 

25. Same with points A (5,9,6), B (5,3,1) and C (5,3,12). 

26. Same with points A (10,1, 4), B (7,10, 4) and C (1, 4, 4). 

27. The V projections of the points A (8, 1, 2), B (10, 3, 8), 
C (4, 3, 10) and D (2,1, 4) form a square. Draw the projections 
and connect them point to point. What are the coordinates of the 
center where AC and BD intersect? 


26 


Engineering Descriptive Geometry 


28. Plot the parallelogram A (11, 3, 3 ), B (3, 3, 3), C (7, 9, 7), 
D (15, 9, 7). The diagonals intersect at E. Give the coordinates 
of E. Describe the ff projector-planes of AB, AC, CD, giving 
lengths of sides of each quadrilateral. Is the plane of the figure 
inclined or oblique? Is AG an inclined or oblique plane? 

29. Plot the quadrilateral 

A (11,10,3), B (3,10,11), C (7,2,7), D (11,4,3). 

Is the plane of the figure horizontal, vertical, inclined or oblique ? 

Is the line AB horizontal, vertical, inclined or oblique ? 

Is the line BC horizontal, vertical, inclined or oblique? 

Is the line CD horizontal, vertical, inclined or oblique? 

Is the line DA horizontal, vertical, inclined or oblique? 


CHAPTER III. 

THE TRUE LENGTH OF A LINE IN SPACE. 

22. The Use of an Auxiliary Plane of Projection. —To find the 
true length of a “ line in space/’ or oblique straight line, an auxil¬ 
iary plane of projection is of great value, and is constantly used 
in all branches of Engineering Drawing. 

A typical solution is shown by Figs. 20 and 21. The essential 
feature is the selection of a. new plane of projection, called an 




auxiliary plane, and denoted by which must be parallel to the 
given line and easily revolved into coincidence with one of the 
regular planes of reference. 

This auxiliary plane is passed parallel to one of the projector- 
planes. In Eig. 20 the plane S' of the cube of planes has been 
replaced by a plane U, parallel to the ff projector-plane, AAnBhB. 
Like that plane, JJ is also perpendicular to H? an ^ the li ne 
of intersection of HJ and JH, is parallel to AnBn. The distance of 
the plane U from the projector-plane may be taken at will and 
in the practical work of drawing it is a matter of convenience, 
choice being governed by the desire to make the resulting figures 
clear and separated from each other. In Fig. 20 the auxiliary 























28 


Engineering Descriptive Geometry 


plane JJ has been established by selecting a point X in H for it 
to pass through. U is an “ inclined plane/' not an “ oblique plane." 

23. Traces of the Auxiliary Plane U-—The auxiliary plane HJ 
cuts the plane V i n a line XN, parallel to the axis of Z. The 
lines of intersection of JJ with the reference planes, are called the 
“ traces 99 of JJ. Since there are three reference planes, there may 
be as many as three traces of HJ. In the case illustrated in Fig. 20, 
there are, however, but two traces. Only one of these three possible 
traces of HJ can be an inclined line. In Fig. 20 the trace XM alone 
is an “ inclined " line. 

We shall see later that the auxiliary plane may be taken per¬ 
pendicular to V or to § as alternative methods. In every case 
there is but one inclined trace, that on the plane to wdiich U is 
perpendicular. It'is this trace which lias the greatest importance in 
the process. For the sake of uniformity, M and N will be assigned 
as the symbols for marking the traces of an auxiliary plane of 
projection. 

24. The U Projectors.— A new system of projectors, AA U , BB U , 
etc., project the line AB upon the plane HJL These projectors, 
being perpendicular distances between a line and a plane parallel to 
it, are all equal, and the projector-plane AA U B U B of Fig. 20 is in 
reality a rectangle. A U B U is therefore equal in length to AB, or 
AB is projected upon U without foreshortening. 

25. Development of the Auxiliary Plane UJ-—The descriptive 
drawing, Fig. 21, is the drawing of practical importance, which is 
based on the perspective diagram, Fig. 20, which shows the mental 
conception of the process employed. In practical work, of course, 
Fig. 21 alone is drawn, and it is constructed by geometrical reason¬ 
ing deduced from the mental process exhibited by Fig. 20. 

In the process of flattening out or “ developing " the nlanes of 
projection, UJ is generally considered as attached or hinged to the 
“ inclined trace," XM in this example. In Fig. 21 {Q has been 
revolved about XM, bringing it into the plane of ff, the trace XN 
having opened out to two lines. N separates into two points and is 
marked N u as a point in UJ and N v as a point in Y, analogous to 
Yh and Y s in the development of the reference planes. The space 
N U XN V , like Y h OY s , may be considered as construction space. 


The True Length of a Line in Space 


29 


26. Fourth Law of Projection—that for Auxiliary Plane, JJ* — 

It will be seen from Fig. 20 that AA h eA v is a rectangle and that 
eA v is equal to A h A. On the descriptive drawing, Fig. 21, these 
two lines, eA v and eA h , form one line perpendicular to OX. This 
is in accordance with the first law of projection of Art. 11. 

As the plane HJ is perpendicular to ff we have the same rela¬ 
tion there, and A Ante A u , Fig. 20, is a rectangle. TcA u is therefore 
equal to A^A, and in the development, Fig. 21, A u k and hAn form 
one line AJcAn perpendicular to XM. 

If from A u and A v , Fig. 20, perpendiculars are let fall upon the 
intersection of IjJ and V (the trace XN) they will meet at the 
common point l, both hA u lX and XlA v e being rectangles. In the 
descriptive drawing, Fig. 21, A u l is perpendicular to XN U , l u l v is 
the arc of a circle, center at X, and l v A v is perpendicular to XN V . 

The following law of projection governs the position of A u in 
the plane U: 

(4) From the regular projections of A draw perpendiculars 
to the traces of HJ- These lines continued into the field 
of U intersect at A u . One of these lines is carried across 
the construction space by the arc of a circle whose center 
is the meeting point of the traces of HJ- 

27. The Graphical Application of this Law to a Point. —The 

procedure for locating the projection A u on the descriptive drawing. 
Fig. 21, after the location of the plane KJ has been determined, is 
as follows: From the adjacent projections of the point draw lines 
perpendicular to the traces of the plane JJ- Continue one of these 
lines across the trace. Swing the foot of the other perpendicular 
to the duplicated trace, and continue it by a line perpendicular to 
this trace to meet the line first mentioned. Their intersection is 
the projection of the point on JJ. In Fig. 21, this requires AiJcA u 
to be drawn perpendicular to XM, and the line A v l v l u A u to be 
traced as shown. , 

28. The True Length of a Line. —The procedure for finding the 
true length of a line consists in first drawing, Fig. 21, a line paral¬ 
lel to one of the projections of the line to act as the trace of the 
auxiliary plane. Where this trace intersects an axis of projection 
perpendicular lines are erected, one perpendicular to the axis, one 


30 


Engineering Descriptive Geometry 


perpendicular to the trace. These lines are the two developed 
positions of the other trace of the plane UJ. Then locate the ex¬ 
tremities of the given line on the auxiliary plane UJ. The line 
joining the extremities is the required projection of the line on UJ> 
and is equal in length to the given line. 

29. Alternative Method of Developing the Auxiliary Plane, KJ.— 
A modification of this construction is shown in the descriptive 
drawing, Eig. 22, in which the plane UJ has been revolved about 
the vertical trace XN until it coincides with the plane V- ATM 
separates into two lines, XMu and XM U . Tc, of Fig. 20, becomes 
Ten and Ay, and the space M\XM U is construction space. A is on 



a horizontal line drawn through A v . Ankh is perpendicular to XMu. 
iciiku is the arc of a circle having X as a center, and h u A u is per¬ 
pendicular to XM U . A u is thus located. 

This method of development of the planes is much less common 
in practical drawing than the other, because, as a rule, it is less 
convenient than the first method. In such cases as occur it offers 
no particular difficulty. Both Figs. 21 and 22 are solutions of 
the problem of finding the true length of the oblique line AB by 
projection on an auxiliary inclined plane, {jj- 

30. Alternative Positions of the Plane UJ.—We saw that the 
exact position of the plane UJ? so long as it remained perpendicular 
to ff and parallel to AnBn, was left to choice governed by practical 









The True Length of a Line in Space 


31 


considerations. JJ itself, however, may be taken perpendicular to 
V and parallel to A V B V , or it may be taken perpendicular to § and 
parallel to A S B S . To get an entire grasp of the subject the student 
is advised to trace Fig. 21 on thin paper, or plot it on coordinate 
paper, points A, B, and X being (6,6,2), (10,10,8) and 
(11, 0, 0), and fold the figure into a paper box diagram, the con¬ 
struction spaces N U XN V and Y%OY s being creased in the middle 
and folded out of the way. Fig. 22 will serve equally well. The 
final result will be a paper box exactly similar to Fig. 20. 

The variation in which JJ is perpendicular to V ma y be plotted, 
passing the new inclined trace of HJ (lettered YM) through the 
point (0, 0, 3) parallel to A V B V . Fold this figure into a paper box, 
the paper being cut along a line YN perpendicular to YM. 

The other variation may be plotted with the inclined trace of U 
on the plane §, parallel to A S B S and passing through the point 
(0,0,6) (f 8 of Fig. 21). Letter this trace Y S M and draw Y S N 
perpendicular to it, inclining up to the right. The paper must be 
cut on this line to enable it to be properly folded. 

31. The Method Applied to a Plane Figure. —The special value 
of this use of the auxiliary plane is seen when one operation serves 
to give the true length of a number of lines at once, and thus 
shows a whole plane figure in its true shape. 

In Fig. 23 the polygon ABODE is shown by its projection, the 
point A alone being lettered. It is noticeable that in V the edges 
all form one straight line. The V projector-planes of the various 
edges are therefore all parts of the same plane, and the polygon 
itself is a plane figure placed perpendicular to V- It may be said 
the polygon is “ seen on edge " in V- 

An auxiliary plane JJ has been taken parallel to the plane of the 
polygon, and therefore perpendicular to V. The trace XM being 
parallel to the V projections of the edges, this auxiliary plane serves 
to show the true length of all the edges at once. The projection on 
U is the true shape of the polygon ABODE. In the case illustrated, 
the HJ projection discloses the fact that the polygon is a regular 
pentagon, a fact not realized from the regular projections, owing 
to the foreshortening to which they are subject. 


32 


Engineering Descriptive Geometry 


This figure is well adapted to tracing and folding into a paper 
box diagram. 

"‘H 




; i 

i 

i 


4 


7 \“ 

i 

4 - 

i 

• 




\ 



Fig. 23. 


32. The True Length of a Line by Revolving About a Projector. 

■—A second method of finding the true length of a line seems in a 
way simpler, but proves to be of much less value in practical work. 
The method consists in supposing an oblique line AB to be revolved 
about a projector of some point in the line until it becomes parallel 
to one of the planes of reference. In this new position it is pro¬ 
jected to the reference plane as of its true length. 

In Fig. 24 the V projector-plane of the line AB has been shaded 
for emphasis (A is the point (1,1, 5), and B is the point (5, 4, 2)). 
The projector AA V has been selected at will, and the V projector- 
plane (of which the line AB is one edge) has been rotated about 
AA V as an axis until it has come into the position A V B' V B'A. In 
its new position, AB' projects to J-J as AhB\. This is the true 
length of the line. During its rotation the point B has moved to 
B f , but in so doing it has not revolved about A as its center, but 
about the point 6 on A V A extended. bA v is equal in length to BB V . 




















The True Length of a Line in Space 


33 


B v moves to B' v , revolving about A v as a center. In Fig. 25, the 
corresponding descriptive drawing, the original projections are 
shown as full lines and the projections of the line after the rota¬ 
tion has occurred are shown by long dashes. 

In V, A V B V swings about A v as a pivot until in its new position 
A V B' V it is parallel to OX. In ff, Bn moves in a line parallel to 
OX (since in Fig. 24 the motion of B takes place entirely in the 
plane of bBB', which is parallel to V)> and as B'n must be verti¬ 
cally above B' v the motion terminates where a line drawn vertically 



up from B' v meets the horizontal line B],B'n. Joining Ah and B \, 
the new H projection is the true length of the given line. The § 
projection is of no interest in this case. The ff and V projections 
of Fig. 25 show the graphical process corresponding to the theory 
of this rotation. In V> B v moves to B' v , whence a vertical pro¬ 
jector meeting a horizontal line of motion from Bn determines B\, 
the new position of Bn . Aj,B'n is the true length of the line. The 
arrow-heads on the broken lines make these steps clear. 

33. Variations in the Method. —The method is subject to wide 
variations. The same projector-plane AA V B V B, Fig. 24, revolving 
about the same projector AA v , might start in the opposite direction 
and swing to a position parallel to §. The graphical process of 
Fig. 25 would then confine itself to V and § instead of V and EU- 




















34 


Engineering Descriptive Geometry 


In addition, tlie rotation might have been about BB V as an axis 
or about the V projector of any point in AB or AB extended. 
Finally, the ff projector-plane or the § projector-plane might 
have been selected and made to revolve into position. There are 
six distinct varieties of the process, each one subject to great modifi¬ 
cations. 

This method can be applied to a plane figure which appears “ on 
edge” in one of the regular views. In Fig. 26 a polygon lies in a 



Fig. 26. 


plane perpendicular to V- There are two varieties of the process 
applicable in this case. Choosing the V projector of the point A 
for the axis of rotation, the whole polygon may be rotated up par¬ 
allel to H, thence its true shape projected upon Jj-fl, or it may be 
revolved down until parallel to §, thence its true shape projected 
upon S- Both methods are shown, though of course in practice one 
at a time should be enough. 

34. A Projector-Plane Used as an Auxiliary Plane.— The two 

processes for finding the true length of a line differ in this respect. 




















The True Length of a Line in Space 


35 


In one the line is projected on a plane which is revolved into 
coincidence with one of the reference planes, by revolving about a 
line in that reference plane. In the second process, a projection 
plane is itself revolved about a projector, that is, about a line 
perpendicular to one reference plane, to a position parallel to a 
second reference plane. The line in its new position is projected 
on the latter plane. 

A method which is a modification of the first process is in many 
cases very simple. A projector-plane is itself used as an auxiliary 
plane, and is revolved into coincidence with the plane to which it is 
perpendicular by rotation about its trace in that plane. In Fig. 23, 
for example, instead of passing XM parallel to A V C V , A V G V would be 
extended to the axis of X, and used itself for the inclined trace of 
the auxiliary, plane. XX would be moved to the right and other 
slight modifications made. 

As in the second method, a projector-plane is here rotated; but 
it is not rotated about a projector, hut about a projection (its 
trace), and the real similarity of the process is with the first 
method, that of the auxiliary plane of projection. It is but a 
special case of this kind. 

In practical drawing, it rarely happens that one of the projector- 
planes can be thus used itself with advantage as an auxiliary plane 
of projection. It leads usually to an overlapping of views and it 
will not be found so useful as the more general method. 

For the continuation of this study, all these methods should be 
at the students’ finger ends. 

35. The True Length of a Line by Constructing a Right Tri¬ 
angle. —These methods of finding the true length of a line are 
generally used for the true lengths of many lines in one operation, 
or for the true shape of a plane figure. When a single line is 
wanted, the construction of a right triangle from lines whose true 
lengths appear on the drawing is sometimes resorted to. In Fig. 24 
the triangle ABb is a right triangle, AB being the hypotenuse and 
AbB the right angle. In the descriptive drawing. Fig. 25, A V B V 
is equal in length to Ab of Fig. 24, and Anb is easily found, equal 
to Ab of Fig. 24. These lines may be laid off at any convenient 


36 


Engineering Descriptive Geometry 


place as the sides of a right triangle, and the hypotenuse measured 
to give the true length of AB. Mathematically the hypotenuse is 

the square root of the sum of the squares of the sides. In the case 

- 2 —- 2 

illustrated A V B V is 5 (itself the square root of A v c“+B v c , or 
V3 2 + 4*) and Anb is 3. The length AB is therefore V5 2 + 3 2 
= V34 = 5.83. 


Problems III. 

(For use with wire-mesh cage, cross-section paper, or blackboard.) 

30. A square in a position similar to the pentagon of Fig. 
26 has the corners A (10,12,2), B (2,12,8), C (2,2,8) and 
D (10, 2, 2). Find its true shape by the use of an auxiliary plane. 

31. A square is in a position similar to the pentagon of Fig. 23. 
The corners are A (9,3,3), B (9,13,3), C (3,13,11), and 
D (3, 3,11). Find its true shape by revolving into a plane par- 
alel to H. 

32. Plot the triangle A (11, 3, 2), B (12, 6,12), C (14,12, 7). 
Find its true shape by the use of an auxiliary plane perpendicular 
to H- 

33. Plot the triangle A (13,14,8), B (10,10,0), C (7,6,8). 
Show the true shape of the triangle by revolving it about AAn 
until in a plane parallel to §. Find the true shape by projection 
on a plane \] perpendicular to ff, whose inclined trace passes 
through the point (0,16, 0). (With the wire-mesh cage turn plane 
S' to serve for this auxiliary plane U-) 

34. Same with triangle A (9, 7, 8), B (12,11,13), C (15,15, 2). 

35. Plot the right triangle A (14, 4, 3), B (14,10, 3), C (6, 4, 9). 
Eevolve it about BB V into a plane parallel to and project its 
true shape on fl-fl. (With the wire-mesh cage put markers at points 
A, B, C and C ', the new position of G.) 

36. Plot the right triangle A (9, 3, 6), B (9, 3, 0), O (15.11, 6). 
Revolve it about AB until in a plane parallel to V an d plot C', the 
new position of the vertex. Revolve it about the same axis into a 
plane parallel to §, and plot C", the new position of the vertex. 
(With the wire-mesh cage put point markers at A, C, C' and C".) 





The True Length of a Line in Space 


37 


37. Plot the square A (14,8,2), B (10, 2, 7J), C (10,14, 7J), 
D (8, 8,12f). The diagonal is 12 units long. Revolve it about 
AA V into a plane parallel to |HI, and project its true shape on ff. 
(With wire-mesh cage put point markers at A, B, C, D } B', C r , 
and D'.) 

38. Plot the triangle A (12,2,14), B (2,2,14), C (7,7,2). 
Revolve it about ABA S into a plane parallel to ff, and project the 
true shape on V* (With wire-mesh cage put markers at points 
A, B, C and A'. On coordinate paper or blackboard show true shape 
by projection on an auxiliary plane \] perpendicular to §, through 
the point (0, 8, 0).) 

(For use on coordinate paper or blackboard, not wire-mesh cage.) 

39. The triangle A (3,7,11), B (13,2,13), C (5,2,1) is a 
triangle in an oblique plane. Find its true shape as follows: BC 
appears at its true length in V- Draw A V D V perpendicular to B V C V - 
AD is an oblique line, but it is perpendicular to BC since its V 
projector-plane AA V D V D is perpendicular to BC. Find the true 
length of AD by any method. On V extend A V D V to E v , making 
D V E V equal to the true length of AD. E V B V C V is the true shape of 
the triangle ABC. 


CHAPTER IV. 

PLANE SURFACES AND THEIR INTERSECTIONS AND 

DEVELOPMENTS. 

36. The Omission of the Subscripts, h, v, and s —In a descriptive 

drawing a point does not itself appear but is represented by its 
projections on the reference planes. This fact has been emphasized 
in the previous chapters. In the more complicated drawings which 
now follow it will save time and will prevent overloading the figures 
with lettering, to omit the subscripts h, v, and 8, and to refer to a 
point and its projections by the same letter. Thus “ A v ” or “ the 
point A in V 99 are expressions which call attention to the projec¬ 
tion of A on but a diagram will show only the letter A at that 
place. If at any time it is necessary to be more precise the sub¬ 
scripts may be restored. They should be used if any confusion is 
caused by their omission. 

If the projections of two points coincide, it is sometimes advis¬ 
able to indicate which point is behind the other in that view by 
forming the letter of fine dots. Referring back to Fig. 14, the 
projections of A and B on § coincide. On this system subscripts 
are omitted and the letter B (on § only) is formed of dots, 
as in Fig. 27. 

37. Intersecting Plane Faces.—Many pieces of machines and 
structures which form the subjects of mechanical drawings, are 
pieces all of whose surfaces are portions of planes, each portion or 
face having a polygonal outline. 

In making such drawings there arise problems as to the exact 
points and lines of intersection, which can be solved by applying 
the laws of projection treated of in the preceding chapters. How 
these intersections are determined from the usual data will now 
be shown. 

38. A Pyramid Cut by a Plane. —As a simple example let us 
suppose that it is required to find where a plane perpendicular to 
V, and inclined at an angle of 30° with ff, intersects an hex- 


Plane Surfaces and Their Intersections 


39 


agonal pyramid with axis perpendicular to ff. Fig. 27 is the 
drawing of the pyramid, having the base ABCDEF and vertex P. 
The cutting plane is an inclined plane such as we have used for an 
auxiliary plane, and its traces are therefore similar to those of an 
auxiliary plane. KL is the inclined trace on V and KIP and LL' 
are the traces parallel to the axis of Y. The problem is to find the 
shape of the polygonal intersection abcdef in ff and g, and its 
true shape. 



The method of solution of all such problems is to take into con¬ 
sideration each edge of the pyramid in turn, and to trace the points 
where they pierce the plane. Thus, the edge PA pierces the given 
plane at a, whose projection on V is first located; for the given 
plane is seen on edge in V? and PA cannot pierce the plane at any 
other point consistent with that condition, a, once located in V> 
can be projected horizontally to the line PA in § and vertically to 
PA in H- 

The true shape of the polygon abcdef may be shown on an aux¬ 
iliary plane, JJ? whose traces are ZM and ZN. In Fig. 27 the 
projection of the pyramid on HJ is incomplete. As it is only to 
show the polygon abcdef the rest of the figure is omitted. 


















40 


Engineering Descriptive Geometry 


39. Intersecting Prisms. —As an example of somewhat greater 
difficulty let it be required to find the intersection of two prisms, 
one, the larger, having a pentagonal base, parallel to ff ? an h the 
other a triangular base, parallel to §. The axes intersect at right 
angles, and the smaller prism pierces the larger. 




Fig. 29. 


The known elements or data of the problem are shown recorded 
as a descriptive drawing in Tig. 28. It shows the projection of 
the jientagon on J-J, of the triangle on and of the axes on V- 
The problem is to complete the drawing to the condition of Fig. 
29, shown on a larger scale. The corners of the pentagonal prism 
are ABODEF and A'B'C'D'E'F' and its axis is PP\ The corners 
of the triangular prism are FGII and F'G'H' and its axis is QQ f . 

40. Points of Intersection. —The general course in solving the 
problem of the intersection of the prisms is to consider each edge 
of each prism in turn, and to trace out where each edge pierces the 
various plane faces of the other prism. When all such points of 





































































Plane Surfaces and Their Intersections 


41 


intersection Pave been determined, they are joined by lines to give 
the complete line of intersection of the prisms. 

To determine where a given edge of one prism cuts a given plane 
face of the other prism, that view in which the given plane face is 
seen as a line only, or is “ seen on edge,” as is said, must be re¬ 
ferred to. Taking the hexagonal prism first, the edges AA', CC', 
and DD' entirely clear the triangular prism, as is disclosed by the 
plan view on ff where they appear “ on end ” or as single points 
only. They, therefore, have no points of intersection with the 
triangular prism and in V and S these lines may be drawn as 
uninterrupted lines, being made full or broken according to the 
rule at the end of Art. 5. BB', as may be seen in JHI, meets the 
small prism. This line when drawn in §, where the plane faces 
FF'G'G and FF'H'H are seen on edge, meets those faces at b and 
F. From § these points are projected to V- The edge BB' con¬ 
sists really of two parts, Bb and b'B'. EE' meets the same two 
faces at points e and e’ determined in the same way. 

FF', when drawn in Jf, is seen to pierce the plane face A A'B'B 
at / and AA'E'E at These points, located in H, are projected 
vertically down to V- GG' in ff pierces BB'C'C at g, and EE'D'D 
at g'. h and li on the line HII' are similarly determined first in 
H and are projected down to V- 

41. Lines of Intersection.—Having found the points of inter¬ 
section of the edges, we determine the lines of intersection of the 
plane surfaces by considering the intersections of plane with plane, 
instead of line with plane. BB' is one line of the plane AA'B'B, 
and pierces the plane FF'G'G (seen on edge in S) at b. b is there¬ 
fore a point of both planes. FF' is a line of the plane FF'G'G, and 
it pierces the plane AA'B'B (seen on edge in fj) at /. f is also a 
point common to both planes. Since these two points are in both 
planes, they are points on the line of intersection of the two planes. 
We therefore connect b and f by a straight line in V? but do not 
extend it beyond either point because the planes are themselves 
limited. 

By the same kind of reasoning b and g are found to be points 
common to BB'C'C and FF'G'G, and are therefore joined by a 
straight line, bg in V. gli also is the line of intersection of two 


42 


Engineering Descriptive Geometry 


planes, and the student should follow for himself the full process 
of reasoning which proves it, e, and g' are points similar to b, f, 
and g. Since the original statement required the triangular prism 
to pierce the pentagonal one, gg', ff, and lull' are joined by broken 
lines representing the concealed portions of the edges GG r , FF ', and 
TIIF of the small prism. Had it been stated that the object was 
one solid piece instead of two pieces, these lines would not exist on 
the descriptive drawing. 

42. Use of an Auxiliary Plane of Projection.—To find the inter¬ 
section of solids composed of plane faces, it is essential to have 



views in which the various plane faces are seen on edge. To obtain 
such views, an auxiliary plane of projection is often needed. 

Fig. 30 shows the data of a problem which requires the auxiliary 
view on U in order to show the side planes of the triangular prism 
“on edge/' (These planes are oblique, not inclined, and therefore 
do not appear “on edge” on any reference plane.) Fig. 31 shows 
the complete solution, the object drawn being one solid piece and 
not one prism piercing another prism, b and d are located by the 
use of the view on JJ. In this case and in many similar cases in 
practical drawing, the complete view on HJ need not be constructed. 





























Plane Surfaces and Tiieir Intersections 


43 


The use of HJ is only to give the position of b and d, which are then 
projected to V- The construction on HJ of the square ends of the 
square prism are quite superfluous, and would be omitted in prac¬ 
tice. In fact, the view UJ would be only partially constructed in 
pencil, and would not appear on the finished drawing in ink.- 

After the method is well understood, there will be no uncertainty 
as to how much to omit. 

43. A Cross-Section.—In practical drawing it often occurs that 
useful information about a piece can be given bv imagining it cut 
by a plane surface, and the shape of this plane intersection drawn. 
In machine drawing, such a section showing only the material 
actually cut by the plane and nothing beyond, is called a “ cross- 



Fig. 32. 


section.” In other branches of drawing other names for the same 
kind of a section are used, The “ contour lines ” of a map are of 
this nature, as well as the “ water lines ” of a hull drawing in 
ISTaval Architecture. 

44. Sectional Views.—The cross-section is used freely in ma¬ 
chinery drawing, but a “ sectional view,” which is a view of a cross- 
section, with all those parts of the piece which lie beyond the plane 
of the section as well, is much more common. 

These sectional views are sometimes made additional to the 
regular views, but often replace them to some extent. Fig. 32 is a 

























































44 


Engineering Descriptive Geometry 


good example. It represents a cast-iron structural piece shown by 
plan, and two sectional views. The laws of projection are not 
altered, but the views bear no relation to each other in one respect. 
One view is of the whole piece, one is of half the piece, and one is 
of three-quarters of it. The amount of the object imagined to be 
cut away and discarded in each view is a matter of independent 
choice. 

In the example the projection on V is a view of half of the 
piece, imagining it to have been cut on a plane shown in ff by the 
line mn. The half between mn and OX has been discarded, and the 
drawing shows the far half. The actual section, the cross-section 
on the line mn, is an imaginary surface, not a true surface of the 
object, and it is made distinctive by “hatching.” This hatching 
is a conventional grouping of lines which show also the material 
of which the piece is formed. For this subject, consult tables of 
standards as given in works on Mechanical Drawing. This pro¬ 
jection on V is not called a “ Front Elevation,” but a “ Front Ele¬ 
vation in Section,” or a “ Section on the Front Elevation.” 

The view projected on S is called a “ Side Elevation, Half in 
Section,” or a “ Half-Section on the Side Elevation.” Since a 
section generally means a sectional view of the object with half 
removed, a half-section means a view of the object with one-quarter 
removed. If, in fl-J, the object is cut by a plane whose trace is np 
and another -whose trace is pq, and the N. E. comer of the object 
is removed, it will correspond to the condition of the object as seen 
in S. 

Sections are usually made on the center lines, or rather on central 
planes of the object. When strengthening ribs or “ webs ” are seen 
in machine parts, it is usual to take the plane of the section just 
in front of the rib rather than to cut a rib or web which lies on the 
central plane itself. This position of the imaginary saw-cut is 
selected rather than the adjacent center line. 

When the plane of a section is not on a center line, or adjacent 
to one, its exact location should be marked in one of the views in 
which it appears “on edge,” and reference letters put at the ex¬ 
tremities. The section is then called the “ section on the line mn.” 

The passing of these section planes causes problems in intersec¬ 
tion to arise, which are similar to those treated in Articles 37-42. 


Plane Surfaces and Their Intersections 


45 


45. Development of a Prism.—It is often desired to show the 
true shape of all the plane faces of a solid object in one view, 
keeping the adjacent faces in contact as much as possible. This 
is called developing the surface on a plane, and is particularly 
useful for all objects made of sheet-metal, as the development forms 
a pattern for cutting the metal, which then requires only to be bent 
into shape and the edges to be joined or soldered. 



Development is a process already applied to the planes of pro¬ 
jection themselves when these planes w^ere revolved about axes until 
all coincided in one plane. The same operation applied to the 
surfaces of the solid itself produces the development. 

The two prisms of Fig. 29 afford good subjects for development. 
Fig. 33 shows the developed surface of the triangular prism, the 
lines g-g and g'-g' showing the lines of intersection with the other 
prism. In this figure it is considered that the surface of the 
triangular prism is cut along the lines GG ', GF, G'F' , GII, and 


















46 


Engineering Descriptive Geometry 


G'lP ; and the four outer planes unfolded, using the lines bound¬ 
ing FF'IPII as axes, until the entire surface is flattened out on the 
plane of FF'H'FL. 

Fig. 34 shows the development of the large prism of Fig. 29, 
with the holes where the triangular prism pierces it when the two 
are assembled. The surface of the prism is cut on the line .4.4', 
and on other lines as needed, and the surfaces are flattened out by 
unfolding on the edges not cut. 

The construction of these developments is simple, since the sur¬ 
faces are all triangles or pentagons whose true shapes are given; 
or are rectangles, the true length of whose edges are already known. 

In Fig. 33 the distances Gg, G'g', Ff, F'f' are taken directly 
from V i n Fig. 29. The points b and e are plotted as follows: 
The perpendicular distance bl to the line GF is taken from V? 
Fig. 29, and Gl is taken from Gl in §, Fig. 29. The other points 
are plotted in the same manner. 

46. Development of a Pyramid.—Fig. 35 shows the development 
of the point of the pyramid, Fig. 27, cut off by the intersecting 



plane whose trace is KL. The base is taken from the projection 
on UJ, where its true shape is given. Each slant side must have its 
true shape determined, either as a whole plane figure (Art. 34), 
or by having all three edges separately determined (Art. 28 or 
Art. 32) In this case Pa and Pd are shown in true length in V\ 
Fig. 27, and it is only necessary to determine the time lengths of 
Pb. and Pc (or their equivalents Pf and Pc) to have at hand all the 
data for laying out the development. The face Pef may be con¬ 
veniently shown in its time shape on an auxiliary plane W? Fig. 27, 
perpendicular to § and cutting S in a trace Y S N as shown. 




Plane Surfaces and Their Intersections 


47 


Problems IV. 

(For use with wire-mesh cage, or on cross-section paper or 

blackboard.) 

40. Plot the projections of the points A (9, 3,16), B (6, 3,16), 
C (6, 8,16), D (9, 18,16), and E (0, 3, 4), F (0, 3, 8), G (0, 8, 8), 
H (0,8,4). Join the projections A to E, B to F, C to G, etc. 
(With wire-mesh cage use stiff wire to represent the lines AE, BE, 
etc.) Show how to find the true shape of every plane surface of 
the figure contained between the 4 lines, and the planes § and ff'. 
On cross-section paper or on blackboard show how to draw the 
development of the surface of the solid. 

41. Same as Problem 40, with points A (10, 8, 0), B (8, 10, 0), 
C (12, 14, 0), D (14,12, 0) on H and E (10, 8, 16), F (6, 12,15), 
G (6,14,15), H (12,10,14) on ]ffl\ 

42. Draw the tetrahedron whose four corners are A (16,2,1.3), 
B (6, 2,13), C (11,14, 13) and D (11,7,1). It is intersected by 
a plane perpendicular to V cutting V in a trace passing through 
the origin, making an angle of 30° with OX. Draw the trace of 
the plane on V- Where are its traces on ff and § ? Show the ff 
and S projections of the line of intersection of the plane and 
tetrahedron. 

43. A solid is in the form of a pyramid whose base is a square 
of 10", and whose height is 8". The comers are A (16,2,10), 
B (10,2,2), C (2,2,8) and D (8,2,16) and the vertex 
E (9,10,9). It is intersected by a plane perpendicular to ff, 
whose trace on ff passes through the origin making an angle of 
30° with OX. Draw the V an d S projections of the intersections 
of the pyramid and plane. Where is the trace of the cutting plane 
on Y ? 

44. A plane is parallel to ff at a distance of 16 inches. A 
square prism has its base in ff, points A (8,2,0), B (3,7,0), 
C (8, 12, 0), D (13, 7, 0). Its other base is in J-J', points A'B'C'D' 
having same x and y coordinates and z coordinates 16. 

A plane S' is parallel to S a distance of 16". A triangular 
prism has its base in S> points E (0,5,8), F (0,13,2), 


48 


Engineering Descriptive Geometry 


G (0,13,14) ; and its other base in S', points E', F', G' having x 
coordinates 16, and y and 0 coordinates unchanged. 

Make the drawing of the intersecting prisms considering the 
triangular prism to be solid and parts of the square prism cut away 
to permit the triangular one to pass through. 

(For use on cross-section jiaper or blackboard, not wire-mesh 
cage.) 

45. A sheet-iron coal chute connects a square port, A (2,4,2), 
B (2,12,2), G (2,4,10), D (2,12,10), with a square hatch, 
E (14, 6, 16), F (14,10,16), G (10,10,16), H (10, 6,16). The 
corners form lines AE, BF, CG, DH and the side plates are bent 
on the lines All and BG. Draw the development of the surface. 

46. Draw the development of the tetrahedron in Problem 42 
with the line of intersection marked on it. 

47. Draw the development of the pyramid in Problem 43 with 
the line of intersection marked on it. 

48. Draw the development of the square prism of Problem 44 
with the line of intersection marked on it. 

49. Draw the development of the triangular prism of Problem 
44 with the line of intersection marked on it. 


CHAPTER Y. 

CURVED LINES. 


47. The Simplest Plane Curve, the Circle.—The geometrical 
natures of the common curves are supposed to be understood. De¬ 
scriptive Geometry treats of the nature of their orthographic pro¬ 
jections. The curves now considered are plane curves, that is, 
every point of the curve lies in the same plane. It is natural, 
therefore, that the relation of the plane of the curve to- the plane of 
projection governs the nature of the projection. 




The simplest plane curve is a circle. Figs. 36 and 37 show the 
three forms in which it projects upon a plane. In Eig. 36, a per¬ 
spective drawing, we have a circle projected upon a parallel plane 
of projection (that in the position customary for V). The pro¬ 
jectors are of equal length and the projection is itself a circle ex¬ 
actly equal to the given circle. 

On a second plane of projection (that in the position of §) per¬ 
pendicular to the plane of the circle the projection is a straight 
line equal in length to the diameter of the circle, AC. The pro¬ 
jectors for this second plane of projection form a projector-plane. 





























50 


Engineering Descriptive Geometry 


In Fig. 37 the circle is in a plane inclined at an angle to the 
plane of projection. The projectors are of varying lengths. There 
must he one diameter of the circle, however, that marked AC, 
which is parallel to the plane of projection. The projectors from 
these points are of equal length, and the diameter AC appears of 
its true length on the projection as A V C V . 

The diameter BD at right angles to AC. has at its extremity B 
the shortest projector, and at the extremity D the longest projector. 
On the projection, BD appears greatly foreshortened as B V D V , 
though still at right angles to the projection of AC and bisected 



The true shape of the projection is an ellipse, of which A V C V is 
the major axis and B V D V is the minor axis. Xo matter at what 
angle the plane of projection lies, the projection of a circle is an 
ellipse whose major axis is equal to the diameter of the circle. 

For convenience the two planes of projection in Fig. 36 have 
been considered as V and §, and the projections lettered accord¬ 
ingly. The plane of projection in Fig. 37 has been treated as if 
it were and the ellipse so lettered. It must be remembered that 
the three forms in which the circle projects upon a plane, as a 
circle, as a line, and as an ellipse, cover all possible cases, and the 
relations between the plane of the circle and the plane of projec¬ 
tion shown in the two figures are intended to be perfectly general 
and not confined to V and S alone. 
















Curved Lines 


51 


48. The Circle in a Horizontal or Vertical Plane.—Passing now 
to the descriptive drawing of a circle, the simplest case is that of 
a circle which lies in a plane parallel to fl, V or §. The projec¬ 
tions are then of the kind shown in Fig. 36, two projections being 
lines and one the true shape of the circle. Fig. 38 shows the case 
for a circle lying in a horizontal plane. The true shape appears in 
HI* The V projection shows the diameter AC, the § projection 
shows the diameter BD. 



49. The Circle in an Inclined Plane.—Fig. 39 shows the circle 
lying in an inclined plane, perpendicular to V? and making an 
angle of 60° with ff. The V projectors, lying in the plane of the 
circle itself, form a projector-plane and the V projection is a 
straight line equal to a diameter of the circle. As the plane of the 
circle is oblique to ff and §, these projections on and S are 
ellipses whose major axes are equal to the diameter of the circle. 
Of course, for any point of the curve, as P, the laws of projection 
hold, as is indicated. The true shape of the curve can he shown by 
















52 


Engineering Descriptive Geometry 


projection on any plane parallel to the plane of the circle. It is 
here shown on the auxiliary plane taken as required. If the 
drawing were presented with projections ff, V an d S? as shown, 
one might at first suspect that it represented an ellipse and not a 
circle; but, if a number of points were plotted on Uj tlm existence 
of a center O' could be proved by actual test with the dividers. 

50. The Circle in an Oblique Plane.—When a circle is in an 
oblique plane, all three projections are ellipses, as in Fig. 40. The 
noticeable feature is that the three major axes are all equal in 
length. 



Fio. 40. 

When an ellipse is in an oblique plane, its three projections are 
also ellipses, hut the major axes will be of unequal lengths. The 
proof of this fact must he left until later. The fact that the three 
projections have their major axes equal must be taken at present as 
sufficient evidence that the curve itself is a circle. 

51. The Ellipse: Approximate Representation.—The ellipse is 
little used as a shape for machine parts. It appears in drawings 
chiefly as the projection of a circle. Some properties of ellipses 
are very useful and should be studied for the sake of reducing the 
labor of executing drawings in which ellipses appear. 

An approximation to a true ellipse by circular arcs, known as the 
“ draftsman’s ellipse/’ may he constructed when the major axis 2 a 
and the minor axis 2 1), Fig. 41, of an ellipse are known. 




Curved Lines 


53 


The steps in the process are shown in Fig. 41. The center of the 
ellipse is at 0. The major axis is AC, equal to 2a. The minor 
axis is DB, equal to 2b. From C, one end of the major axis, lay 
off CE, equal to b. The point E is at a distance equal to a — b from 
0 and at a distance equal to 2a—b from A. This last distance is 
the radius of a circular arc which is used to approximate to the 
flat sides of the ellipse. It may he called the “side arc.” Setting 
the compass to .the distance AE and using I) and B as centers, 
points II and G are marked on the minor axis, extended, for use 
as centers for the “ side arcs.” These arcs are now drawn (passing 
through the points D and B ), as shown in the 2nd stage of the 
process. 



Fig. 41. 

By nse of the bow spacer, the distance OE is bisected and the 
half added to itself, giving the point F (distant f (a — b) from 0). 
F is the center of a circular arc which approximates to the end of 
the true ellipse. With F as center, and FC as radius, describe this 
arc. If this work is accurate, this u end arc ” will prove to be 
tangent to the side arcs already drawn, as shown in the 3rd stage 
of the process. If desired, the exact point of tangency of the two 
arcs, K, may be found by joining the centers II and F and extend¬ 
ing the line to K. F is swung about 0 as center by compass or 
dividers to F', for the center of the other “ end arc.” In inking 
such an ellipse, the arcs must be terminated exactly at the points 
of tangency, K and the three similar points. 

This method is remarkably accurate for ellipses whose minor 
5 























54 


Engineering Descriptive Geometry 


axes are at least two-thirds the length of their major axes. It 
should always be used for such wide ellipses, and if the character 
of the drawing does not require great accuracy, it may he used 
even when the minor axis is but half the length of the major axis. 
For all narrow ellipses, exact methods of plotting should be used. 

52. The Ellipse: Exact Representation.—The true and accurate 
methods of plotting an ellipse are shown in Figs. 42, 43, and 44. 
Fig. 42 is a convenient method when the major axis AC and minor 
axis BD are given, bisecting each other at 0. Describe circles with 
centers ot 0, and with diameters equal to AC and BD. From 0 
draw any radial line. From the point where this radial line meets 
the larger circle draw a vertical line, and from the point where it 
cuts the smaller circle draw a horizontal line. Where these lines 



meet at P is located a point on the ellipse. By passing a large 
number of such radial lines sufficient points may be found between 
D and C to fully determine the quadrant of the ellipse. Having 
determined one quadrant, it is generallv possible to transfer the 
curve by the pearwood curves with less labor than to plot each 
quadrant. 

With the same data a second method, Fig. 43, is more convenient 
for work on a large scale when the T-square, beam compass, etc., 
are not available. 

Construct a rectangle using the given major and minor axes as 
center lines. Divide DE into any number of equal parts (as here 
shown, 4 parts), and join these points of division with C. Divide 
DO into the same number of equal parts (here, 4). From A 
draw lines through these last points of division, extending them to 
the first system of lines intersecting the first of the one system with 


















Curved Lines 


00 


the first of the other, the second with the second, etc. These inter¬ 
sections, 1, 2, 3, are points on the ellipse. 

The third method, an extension or generalization of the second, is 
very useful when' an ellipse is to be inscribed in a parallelogram, the 
major and minor axes being unknown in direction and magnitude. 
Lettering the parallelogram A'B'C'D' in a manner similar to the 
lettering in Fig. 43, the method is exactly the same as before, D'E' 
and D'O being divided into an equal number of parts and the lines 
drawn from C' and A'. The actual major and minor axes, indicated 
in the figure, are not determined in any manner by this process. 

53. The Helix.—The curve in space (not a plane curve) which 
is most commonly used in machinery, is the helix. This curve is 
described by a point revolving uniformly about an axis and at the 
same time moving uniformly in the direction of that axis. It is 
popularly called a “ cork-screw ” curve, or “ screw thread,” or even, 
quite incorrectly, a “ spiral.” 

The helix lies entirely on the surface of a cylinder, the radius of 
the cylinder being the distance of the point from the axis of rota¬ 
tion, and the axis of the cylinder the given direction. 

Fig. 45 represents a cylinder on the surface of which a moving 
point has described a helix. Starting at the top of the cylinder, at 
the point marked 0, the point has moved uniformly completely 
around the cylinder at the same time that it has moved the length 
of the cylinder at a uniform rate. The circumference of the top 
circle of the cylinder has been divided into twelve equal parts by 
radii at angles of 30°, the apparent inequality of the angles being due 
to the perspective of the drawing. The points of division are marked 
from 0 to 11, point 12 not being numbered, as it coincides with 
point 0. The length of the cylinder is .divided into twelve equal 
parts on the vertical line showing .the numbers from 0 to 12, and 
at each point of division a circle, parallel to the top base, is de¬ 
scribed about the cylinder. The helix is the curve shown by a 
heavy line. From point 0, which is the zero point of both move¬ 
ments, the first twelfth part of the motion carries the point from 0 
to 1 around the circumference, and from 0 to 1 axially downward, 
at the same time. The true movement is diagonally across the 
curved rectangle to the point marked 1 on the helix. This move- 


56 


Engineering Descriptive Geometry 


ment is continued step by step to the points 2, 3, etc. In the posi¬ 
tion chosen in Eig. 45, points 0, 1, 2, 3, 4, 12 are in full view, 
points 5 and 11 are on the extreme edges, and the intermediate 



points (from 6 to 10) are on the far side of the cylinder. The 
construction lines for these latter points have been omitted, in order 
to keep the figure clear. 








































































Curved Lines 


57 


54. Projections of the Helix.—The projection of this curve on a 
plane parallel to the axis of the cylinder is shown to the left. The 
circles described about the cylinder become equidistant parallel 
straight lines. The axial lines remain straight but are no longer 
equally spaced, and the curve is a kind of continuous diagonal to 
the small rectangles formed by these lines on the plane of projec¬ 
tion. . 

The projection of the helix on any plane perpendicular to the 
axis of the cylinder is a circle coinciding with the projection of the 
cylinder itself. The top base is such a plane and on it the projec¬ 
tion of the helix coincides with the circumference of the base. 

55. Descriptive Drawing of the Helix.—The typical descriptive 
drawing of a helix is shown in Fig. 4G. The axis of the cylinder is 
perpendicular to ff, and the top base is parallel to ff. The helix 
in H appears as a circle. In V it appears as on the plane of pro¬ 
jection in Fig. 45, but this view is no longer seen obliquely as is 
there represented. 

This V projection of the helix is a plane curve of such import¬ 
ance as to receive a separate name. It is called the “ sinusoid.” 
Since the motion of the describing point is not limited to one com- 
plete revolution, it may continue indefinitely. The part drawn is 
one complete portion and any addition is but the repetition of the 
same moved along the axial length of the curve. The proportions 
of the curve may vary between wide limits depending on the rela¬ 
tive size of the radius of the cylinder to the axial movement for one 
revolution. This axial distance is known as the “ pitch ” of the 
helix. 

In Fig. 46 the pitch is about three times the radius of the helix. 
In Fig. 47, a short-pitch helix is represented, the pitch being about 
j the radius, and a number of complete rotations being shown. 

The proportions of the helix depends therefore on the radius and 
on the pitch. To execute a drawing, such as Fig. 46, describe first 
the view of the helix which is a circle. Divide the circumference 
into any number of equal parts (12 or 24 usually). From these 
points of division project lines to the other view or views. Divide 
the pitch into the same number of equal parts, and draw lines per¬ 
pendicular to those already drawn. Pass a smooth curve through 


58 


Engineering Descriptive Geometry 


the points of intersection of these lines, forming the continuous 
diagonal. In Figs. 45 and 46 the helix is a right-hand helix/’ 
The upper part of Fig. 47 shows a left-hand helix, the motion of 
rotation being reversed, or from 12 to 11 to 10, etc. The ordinary 



screw thread used in machinery is a very short-pitched right-hand 
helix. It is so short indeed that it is customary to represent the 
curve by a straight line passing through those points which would 
be given if the construction were reduced to dividing the circum- 

















































































































































































































Curved Lines 


59 


ference and the pitch into 2 equal parts. This is shown in the 

lower part of Fig. 47, where only the points 0, 6 and 12 have been 
used. 

The concealed portion of the helix is then omitted entirely, no 
broken line for the hidden part being allowed by good practice. 

56. The Curved Line in Space.—A curve in space may some¬ 
times be required, one which follows no known mathematical law, 
but which passes through certain points given by their coordinates. 
For example, in Fig. 48, four points, A (12,1,9), B (5,4,6), 



Fig. 48. 


C (2,4,4) and 1) (2,5,1), were taken as given and a “ smooth 
curve, the most natural and easy curve possible, has been passed 
through them. It is fairly easy to pass smooth curves through the 
projections of the 4 points on each reference plane, but it is essen- 

3 d. tli original points obey the laws of pro¬ 
jection of Art. 11, but every intermediate point as well. The 
views must check therefore point by point and the process of trac¬ 
ing the curve must be carried out about as follows: The projec¬ 
tions of the 4 points on V and § are seen to be more evenly ex¬ 
tended than those on J-fl, and smooth curves are made to pass 
through them by careful fitting with the draftsman's curves. The 



























60 


Engineering Descriptive Geometry 


view on § cannot now be put in at random, but must be constructed 
to correspond to the other views. To fill in the wide gap between 
An and Bn an intermediate point is taken, as E v on A V B V . By a 
horizontal line E s is defined. From E v and E s the Ji projection 
(En) is plotted by the regular method of checking the projections 
of a point. As many such intermediate points may be taken as may 
seem necessary in each case. 

To define the sharp turn on the curve between Cn and Dn, one 
or more extra points, as Fj„ should be plotted from the V and S 
projections. Thus every poorly defined part is made definite and 
the views of the line mutually check. The work of “ laying out ” 
the lines of a ship on the “ mold-loft floor ” of a shipbuilding plant 
is of this kind, with the exception that the curves are chiefly plane 
curves, not curves in space. 


Problems V. 

(For blackboard or cross-section paper.) 

50. Make the descriptive drawing of a circle lying in a plane 
parallel to S? center at C (3, 6, 7) and radius 5. 

51. Make the descriptive drawing of a circle lying in a plane 
perpendicular to V> making an angle of 45° with H (the trace in 
H passing through the points (18,0,0), and (0,0,18)). The 
center of the circle is at C (9, 6, 9), and the radius is 5. (Make 
the V projection first, then a projection on an auxiliary plane UJ. 
From these views construct the ff and § projections, using 8 or 9 
points. 

52. Make the descriptive drawing of a circle in a plane perpen¬ 
dicular to H, the trace in ff passing through the points (12, 0, 0) 
and (0,16,0). The center is at (6,8,10) and the radius 8. 
(Draw plan and auxiliary view showing true shape first, and from 
those views construct projections on V and §.) 

53. An ellipse lies in a plane passing through the axis of Y, 
making angles of 45° with H and §. The ff projection is a circle, 
center at (10,10, 0) and radius 8. Prove that the § projection is 
also a circle and find the true shape of the ellipse by revolving the 
plane of the ellipse into the plane ff. 


Curved Lines 


61 


54. An ellipse lies in a plane passing through the axis of Y, 
making an angle of 60° with ff and 30° with g. The H projec¬ 
tion is a circle, center at (8, 8, 0), radius 6. Find the true shape 
of the ellipse. Construct the view on g by projecting points for 
center and for the extremities of the axes of the ellipse. Pass a 
draftsman’s ellipse through those points. Show that no appreci¬ 
able error can be observed. 

55. Construct a draftsman’s ellipse, on accurate cross-section or 
coordinate paper, with major axis 24 units, and minor axis 12 units. 
Perform the accurate plotting of the true ellipse on the same axes 
by the method of Fig. 43, using 6 divisions for BE and EC. Note 
the degree of accuracy of the approximate process. 

56. On coordinate paper, plot an ellipse by the method of Fig. 
43, the major axis being 16 units long and the minor axis 8 units. 
Plot another ellipse whose major axis is 18 and whose minor axis 
is 12. (To divide the semi-minor axis of 6 units into 4 equal parts, 
use points of division on the vertical line CE instead of OB. CE 
being twice as far from A as OB, 12 units must be used for the 
whole length, and these divided into 4 parts.) 

57. On isometric paper pick out a rhombus like the top of Fig. 
19, but having 8 units on each side. Inscribe an ellipse by plotting 
by the method of Fig. 44. 

58. Make the descriptive drawing of a helix whose axis is per¬ 
pendicular to g through the point (0,7,7). The pitch of the 
helix is 12, and the initial point is (2, 5, 2). Draw the ff and V 
projections of a right-hand helix, numbering the points in logical 
order. 

59. Connect the 4 points A (10, 8,10), B (8, 10, 6), C (6, 9, 4) 
and B (2, 2, 4) by a smooth curve, filling out poorly defined por¬ 
tions in g from the ff and V projections. 


CHAPTER VI. 

CURVED SURFACES AND THEIR ELEMENTS. 


57. Lines Representing Curved Surfaces.—To represent solids 
having curved surfaces, it is not enough to represent the actual 
corners or edges only. Hitherto only edges have appeared on de¬ 
scriptive drawings, and it has been a feature of the drawings that 
every point represented on one projection must be represented on 
the other projections, the relation between projections being strictly 
according to rule. We now come to a class of lines which do not 
appear on all three views, lines due to the curvature of the surfaces. 

The general principle, called the “ Principle of Tangent Projec¬ 
tors,” governing this new class of lines is as follows: In projecting 
a curved surface to a given plane of projection (by perpendicular 
projectors, of course) all points, and only those points, whose pro¬ 
jectors are tangent to the curved surface should be projected. A 
good illustration of this principle is shown in Fig. 45, where the 
cylinder is projected upon the plane of projection. The top and 
bottom bases are edges, and project under the ordinary rules, but 
along the straight line 0, 1,2, . . . ., 12 the curved surface of the 
cylinder is itself perpendicular to the plane of projection. If from 
any point on this line a projector is drawn to the plane of projection 
(as is shown in the figure for the points 1, 2, 3, etc.), this projector 
is tangent to the cylinder. The whole line therefore projects to 
the plane of projection. The projection of the cylinder on a plane 
parallel to its axis is therefore a rectangle, two of its sides repre¬ 
senting the circular bases and the two other edges representing the 
curved sides of the cylinder. 

58. The Right Circular Cylinder.—The complete descriptive 

drawing of a cylinder is therefore as shown in Fig. 49. This cylin¬ 
der is a right circular cylinder. Mathematicians consider that the 
cylinder is “ generated ” bv revolving the line A A! about PP\ the 
axis of the cylinder. The generating line in any particular posi- 


Curved Surfaces and Their Elements 


63 


t.ion is called an “ element ” of the surface. Thus AA', BB', CC, 
etc., are elements. 

When the cylinder is projected upon Y, AA ' and CC r are the 
elements which appear in V because the V projectors of all points 
along those lines are tangent to the cylinder, as can be seen from 
the view on ff. The elements which are represented by lines on 
§ are BB' and DD '. 

The right circular cylinder may also be considered as generated 
by moving a circle along an axis perpendicular to its own plane 
through its center. 



In Fig. 45 consider the top base of the cylinder to be moved 
down the cylinder. Each successive position of the circle is a “ cir¬ 
cular element” of the cylinder. The circles through the points 
1, 2, 3, etc., are simply circular elements of the cylinder taken at 
equal distances apart. 

59. The Inclined Circular Cylinder.—Fig. 50 shows an inclined 
circular cylinder . It has circular and straight line elements as 
before, though it cannot he generated bv revolving a line about 
another at a fixed distance, but can be generated by moving the 
circle ABCD obliquely to A'B'C'D ', the center moving on the axis 
PP'. The straight elements are all parallel to the axis. The cross- 
section of a cylinder is a section taken perpendicular to the axis. 














































64 


Engineering Descriptive Geometry 


In this case the cross-section is an ellipse, and for this reason the 
Inclined Circular Cylinder is sometimes called the Elliptical Cyl¬ 
inder. 

60. Straight and Inclined Circular Cones. —If a generating line 
AP, Fig. 51, meets an axis PP' at a point P, and is revolved about 
it, it will generate a Straight Circular Cone. The cone has both 
straight and circular elements, the circular elements increasing in 
size as they recede from the vertex P. The base ABCD is one of 
the elements. 



The Inclined Circular Cone (Fig. 52) has straight and circular 
elements, but it is not generated by revolving a line about the axis. 
The circular elements move obliquely along the axis PP' and in¬ 
crease uniformly as they recede from the vertex P. 

61. The Sphere. —The Sphere can be generated by revolving a 
semicircle about a diameter. Each point generates a circle, the 
radii of the circles for successive points having values varying 
between 0 and the radius of the sphere. Since the sphere can be 
generated by using any diameter as an axis, the number of ways in 
which the surface can be divided into circular elements is infinite. 

62. Surfaces of Revolution. —In general, any line, straight or 
curved, may be revolved about an axis, thus creating a surface of 
revolution. Every point on the “ generating line” creates a “cir- 
































Curved Surfaces and Their Elements 


65 


cular element ” of the surface, and the plane of each circular ele¬ 
ment is perpendicular to the axis of the surface. 

The straight circular cylinder is a simple case of the general 
class of surfaces of revolution. To generate it a straight line is 
revolved about a parallel straight line. The different points of the 
generating line create the circular elements of the cylinder, and 



Fig. 53. 



the different positions of the generating line mark the straight ele¬ 
ments. The cone and the sphere are also surfaces of revolution, as 
they are generated by revolving a line about an axis. 

If a circle be revolved about an axis in its own plane, but en¬ 
tirely exterior to the circle, a solid, called an “ anchor ring,” is 
generated. A small portion of this surface, part of its inner surface, 
is often spoken of as a “ bell-shaped surface,” from its similarity 
to the flaring edge of a bell. 

Any curved line may create a surface of revolution, but in de- 


































66 Engineering Descriptive Geometry 

signs of machinery lines made np of parts of circles and straight 
lines are most frequently used. Figs. 53 and 54 show tv r o exam¬ 
ples which illustrate v r ell the application of the Principle of Tan¬ 
gent Projectors. The generating line is emphasized and the cen¬ 
ters of the various arcs are marked. 

Any angular point on the generating line, as a (Pig. 53), creates 
a circular edge on the surface. This edge appears as a circle on the 
plan (as aa' on H), and as a straight line, equal to the diameter, 
on the elevation (as aa! on V). See also the point li (Fig. 54). 
In addition, any portion of the generating line which is perpen¬ 
dicular to the axis, as b (Fig. 53), even if for an infinitely short 
distance only,, creates a line on the side view, as bb' on V? but no 
corresponding circle on H. A V projector from any point on the 
circular element created by the point b is tangent to the surface, 
and therefore creates a point on the drawing, hut an H projector 
is not tangent to the surface, e is a similar point, and so also is 
j of Fig. 54. 

Any point, as c, Fig. 53, where the generating line is parallel to 
the axis for a finite, or for an infinitely small distance, generates 
a circular element, from ever}' point of which the ff projectors are 
tangent to the surface, but the V projectors are not. A circle cc' 
appears, therefore, on the plan for this element of the surface of 
revolution, but no straight line on the side view, d is a similar 
point, as are also f and a, on Fig. 54.* 

63. The Helicoidal Surface.—If a line, straight or curved, is 
made to revolve uniformlv about an axis and move uniformly along- 
the axis at the same time, every point in the line will generate a 
helix of the same pitch. The surface swept up is called a Ileli- 
coidal Surface. 

The generating line chosen is usually a straight line intersecting 
the axis. The surfaces used for screw threads are nearly all of 
this kind. Fig. 55 gives an example of a sharp A-threaded screw, 
the two surfaces of the thread having been generated by lines in¬ 
clined at an angle of 60° to the axis. Fig. 56 shows a square 
thread, the generating lines of the two helicoidal surfaces being 
perpendicular to the axis. Any particular position of the straight 
line is a “ straight element ” of the helicoidal surface. 


Curved Surfaces and Their Elements 


67 


64. Elementary Intersections. —In executing drawings of ma¬ 
chinery it is often necessary to determine the line of intersection of 
two surfaces, plane or curved. The simplest lines of intersection 
are such as coincide with elements of a curved surface. They may 



be called “ Elementary Intersections." An elementary intersection 
may arise when a curved surface is intersected by a plane, so placed 
as to bear some simple relation to the surface itself. 

In Fig. 49, any plane perpendicular to the axis of the cylinder 
intersects it in a circular element of the cylinder, and any plane 
parallel to the axis of the cylinder (or containing it) intersects 




















































68 


Engineering Descriptive Geometry 


it (if it intersects it at all) in two straight line elements of the 
cylinder. 

In Fig. 50 any plane parallel to the base of the cylinder inter¬ 
sects it in a circular element, and any plane parallel to the axis, 
or containing it, intersects it in straight elements of the cylinder. 

In Fig. 51 or 52 any plane parallel to the base of the cone inter¬ 
sects it in a circular element, and any plane containing the vertex 
of the cone (if it intersects at all) intersects the cone in straight 
elements. 

In Fig. 53 or 54 any plane perpendicular to the axis of the sur¬ 
face of revolution intersects it in a circular element. 

In Fig. 55 or 56 any plane containing the axis of the screw inter¬ 
sects the helicoidal surfaces in straight elements. The plane per¬ 
pendicular to H, cutting H in a trace PQ, and cutting V in. a 
trace QU, cuts the helicoidal surfaces at each convolution in straight 
elements. Only ab and ab are marked on the figure. 

J o 

Problems VI. 

(For blackboard or cross-section paper or wire-mesh cage.) 

60. Draw the projections of a cylinder whose axis is P (6, 2, 6), 
P' (6, 16, 6), and radius 5. Draw the intersection of this cylinder 
with a plane parallel to H, at 4 units from [rj, and with a plane 
parallel to Y, 10 units from V. 

61. An inclined circular cylinder has its bases parallel to g. Its 
axis is P (2, 7, 7), P' (14, 7,13). Its radius is 5. Draw the V 
and § projections and the intersections with a plane parallel to g, 
6 units from g, and with a plane parallel to V, 3 units from V* 

62. Draw a cone with vertex at P (4,8,8), center of base at 
P' (16,8,8), and radius 6, base-line in a plane parallel to §. 
Draw the intersection with a plane parallel to g, 12 units from g, 
and with a plane perpendicular to g, whose trace in § passes 
through the points (0, 8, 8) and (0,14, 0). 

63. An oblique cone has its vertex at P (16, 8, 4), its base in a 
plane parallel to H, center at P' (8,8,16), and radius 5. Draw 
the intersection with a plane parallel to H, 12 units from fffl, and 
with a plane containing the axis and the point (16, 0,16). 


Curved Surfaces and Their Elements 


69 


64. A cone has an axis P (8, 2, 2), P' (8,14,10). Its base is in 
a plane parallel to V, 10 units from V and its radius is 6 units. 
Draw the intersection with a plane containing the vertex and the 
points (0, 14, 12) and (16, 14, 12). 

65. A surface of revolution is formed by revolving a circle whose 
center is at (12, 8, 8) and radius 3 units, lying in a plane parallel 
to Y, about an axis perpendicular to Ji at the point (8, 8, 0). It 
is cut by a plane parallel to fj at a distance of 6 units from ff. 
Draw the intersections. 

66. A sphere lias its center at (8, 8, 9) and radius 5 units. 
Draw the intersection with a cylinder whose axis is P (8, 8, 0), 
P' (8,8,16), and whose radius is 4 units, its bases being planes 
perpendicular to its axis. 

67. A sphere has its center at (8, 8, 8) and radius 5 units. Find 
its intersection with a cone whose vertex is P (0, 8, 9), center of 
base is (16, 8, 8 ), and radius of base 6 units, the base being in a 
plane §' parallel to §. 

68. In Fig. 53 let the generating line Pabcde be revolved about 
ee' as an axis. Assume any dimension for the line and draw the 

V and § projections of the surface of revolution thus formed. 
Draw the intersection with a plane parallel to § just to the right 
of d. 

69. In Fig. 54 let the generating line Pfgh be revolved about 
hid as an axis. Assume any dimensions for the line and draw the 

V and S projections of the surface of revolution formed. 


6 




CHAPTER VII. 

INTERSECTIONS OF CURVED SURFACES. 


65. The Method of the Intersection of the Intersections.—The 

determination of the line of intersection of two curved surfaces (or 
of a curved surface and a plane), when not an “ Elementary Inter¬ 
section,” is of much greater difficulty and requires a clear under¬ 
standing of the nature of the curved surfaces themselves, and some 
little ingenuity in applying general principles. 

The method generally relied upon for the solution is the use of 
auxiliary intersecting planes so chosen as to cut elementary inter¬ 
sections with each of the given surfaces. These elementary inter- 
sections are drawn and the points of intersection of the intersec¬ 
tions are identified and recorded as points on the required line of 
intersection. This method is spoken of as “ finding the intersec¬ 
tion of the intersections.” When a number of auxiliary planes 
have been used in this way, a smooth curve is passed through the 
points on the required intersection of the surfaces, as described in 
Art. 55. It should not be necessary, however, to interpolate points 
to fill out gaps as was done in Fig. 48 for E and F. This can be 
done better by the use of more auxiliary intersecting planes. Ex¬ 
amples of this method will make it clear. 

66. An Inclined Circular Cylinder Cut by an Inclined Plane.— 
In Eig. 57 an inclined cylinder, axis PP', is cut by a plane perpen¬ 
dicular to V, and inclined to j-J. The traces of this plane are IJ 
in H, JK in V* and KL in §. 

It is an Inclined Plane (see Art. 19), not an Oblique Plane. 
Having the descriptive drawing of the cylinder and the traces of 
the plane given, the problem is to draw the line of intersection of 
the surfaces. It is well-known that in this case the line of inter¬ 
section is an ellipse, but the method of determining it permits the 
ellipse to be plotted whether it is recognized as such or not. No 
use is to be made of previous knowledge of the nature of the curve 


Intersections of Curved Surfaces 71 

of intersection of any of the cases treated in this and the next 
chapter. 

Two variations of the method are applicable in this case. In the 
first method, auxiliary intersecting planes may be taken 'parallel to 
the axis of the cylinder. The simplest method of doing this is to 
take auxiliary planes parallel to V? since the axis itself is parallel 



to V* Let R'R be the trace on H, and RR" the trace on § of a 
plane parallel to V. We may call this plane simply “ R.” 


Let e and f be the points where R'R cuts the top base of the cyl¬ 
inder. Project these points from H to V anc ^ in V draw ee' and 
ff parallel to PP'. These straight elements of the cylinder are the 
lines of intersection of the auxiliary plane with the cylinder. As 
a check on the work, e' and f, where R'R in H cuts the bottom 
base of the cylinder, should project vertically to e' and f in V- 













































72 


Engineering Descriptive Geometry 


The auxiliary plane cuts the given plane JK in a line of inter¬ 
section whose projection on V coincides with JK itself. 

The points j and k, where ec' and ff' intersect JK, are the “ inter¬ 
sections of the intersectionsf’ and are therefore points on the line 
of intersection of the cylinder and the plane K. Project j and k 
to R'R on ff and to RR" on g. These are points on the required 
curves in ff and §. By extending in ff the projecting lines of 
j and k as far above the axis PP' as j and Jc are below it, f and k', 
points on the upper half of the cylinder, symmetrical with j and k 
on the lower half, are found. The construction is equivalent to 
passing a second auxiliary plane parallel to PP' at the same dis¬ 
tance from PP' as R, but on the other side. 

By passing a number of planes similar to R, a sufficient number 
of points are located to define accurately the ellipse abed in ff 
and §. 

The true shape of this ellipse is shown in HJ, a plane parallel to 
JK, at any convenient distance. In the example chosen, the plane 
JK has been taken perpendicular to PP', so that the ellipse abed is 
the true cross-section of the cylinder. Nothing in the method de¬ 
pends on this fact and it is perfectly general and applicable to any 
inclined plane. 

A variation may be made by passing the auxiliary planes per¬ 
pendicular to V and parallel to PP '. ee' in V may be taken as the 
trace of such a plane. The intersections of this auxiliary with both 
surfaces should be traced and the intersection of the intersections 
identified and recorded as a point of the curve required, j and f 
are the points thus found. This method indeed requires the same 
construction lines as before, but gives a different explanation to 
them. 

67. A Second Method Using Circular Elements of the Cylinder.— 

A plane parallel to the base of the cylinder and therefore, in this 
case, parallel to ff, will cut the cylinder in a line of intersection 
which is one of the circular elements of the cylinder. Let T'T and 
TT", in Fig. 58, be the traces of a plane " T” parallel to ff. The 
axis of the cylinder PP' pierces the plane T at p. p is therefore 
the center of the circle of intersection of the auxiliary plane T with 
the cylinder. Project p to ff, and using p as a center and with a 
radius equal to pt, describe the circle as shown. 


Intersections of Curved Surfaces 


73 


The planes T and JK are both perpendicular to V or “ seen on 
edge ” in V- Their line of intersection is therefore perpendicular 
to V, or is “ seen on end 99 in Y, as the point j. Project j to ff, 
where it appears as the line jj'. This line is the intersection of the 
two planes. 

The points j and j' 9 where this line of intersection jj' meets the 
circular intersection whose center is at p, are the “ intersections of 
the intersections/' and are points on the required curve. 



Planes like T, at various heights on the cylinder, determine pairs 
of points on the curve of intersection on ff. From H and V the 
points may be plotted on § by the usual rules of projection, thus 
completing the solution. 

68. Singular or Critical Points. —It is nearly always found that 
one or two points on the line of intersection may be projected di¬ 
rectly from some one view to the others without new construction 
lines. In this case a and c in V> Tig. 57, may be projected at once 



















































74 


Engineering Descriptive Geometry 


to H and §. They correspond theoretically to points determined 
by a central plane, cutting Jil in a trace PP'. b and d may also be 
projected directly, as they correspond to planes whose traces in 
fi are BB' and DIP. These critical points should always be the 
first points identified and recorded, though usually no explanation 
will be given, as they should be obvious to any one who lias grasped 
the general method. 

69. A Cone Intersected by an Inclined Plane.—Eig. 59 shows 



the descriptive drawing of a right circular cone intersected by an 
inclined plane whose traces are JK and KL. Two methods of solu¬ 
tion are shown. 

A plane R, containing the axis PP', and therefore perpendicular 
to H, is shown by its traces R'R and RR". It intersects the cone 
in the elements Pj and Pk. From ff project these points j and lc 
to Y, and draw the elements in V- The V projection of the inter¬ 
section of R with the plane JK is the line JK, and the points e and 
f are the intersections of the intersections, e and f are now pro- 








































Intersections of Curved Surfaces 


jected to the plan, where they necessarily lie on the line RR". Sym¬ 
metrical points e' and f are also plotted and all four points trans¬ 
ferred to the side elevation. 

A plane T perpendicular to the axis PP' whose traces are T'T 
and TT" may be used instead of R. Its intersection with the cone 
is a circle, seen on edge in the front elevation as the line Till'. Its 
center is g, and radius is gh. Draw this circle in the plan. The 



intersection of T with the plane JK is a line, seen on end, as the 
point f of the front elevation. Draw ff in ff as this line. The 
points f and f are the intersections of the intersections. 

70. Intersection of Two Cylinders.—Fig. 60 shows the inter¬ 
section of two cylinders. Since they are right cylinders, and their 
axes are at right angles, planes parallel to any one of the three 
reference planes will cut only straight or circular elements of the 
cylinders. By the solution, Fig. 60, auxiliary planes parallel to V 



































































76 


Engineering Descriptive Geometry 


have been chosen, the traces of one being R'R and RR ". This plane 
intersects the vertical cylinder in the lines hk' and IV, and it inter¬ 
sects the horizontal cylinder in the lines mm! and nn. The inter¬ 
sections of these intersections are the points marked r. 

If the axes of the cylinders do not meet but pass at right angles, 
no new complication is introduced. If the axes of the cylinders 
meet at an angle, and one or both cylinders are inclined, the choice 



of methods may be greatly reduced, but one method is always pos¬ 
sible. To discover it, try planes parallel to the axes of both cylin¬ 
ders, or parallel to one axis and to one plane of reference; or in 
some manner bearing a definite relation to the nature of the sur¬ 
faces. 

71. Intersection of a Cylinder and a Sphere. —In Fig. 61 a 
sphere is intersected by a cylinder, whose axis PP' does not pass 
through the center of the sphere at Q. In the solution, Fig. 61, 





























































Intersections oe Curved Surfaces 


77 


auxiliary planes parallel to V have been chosen, the traces of one of 
them being B'B and BE”. The plane B cuts the sphere in a circle 
whose diameter is eg, as given by the plan. This circle is described 
in V. The intersections of this circle with the elements of the 
cylinder 1:1:' and IV are the points marked r, points on the required 
curve of intersection. 

In this case the points are first determined on the front elevation 
and then projected to the side elevation. Solutions by planes par¬ 
allel to fl-fl or to § may be made, requiring however different con¬ 
struction lines. 



72. Intersection of a Cone and a Cylinder: Axes Intersecting.— 

In Fig. 62 a cone and a cylinder intersect at right angles. The 
solution chosen is by horizontal planes, as T. 

An alternate solution is by planes perpendicular to §, and con¬ 
taining the point P. The planes must cut both surfaces, and their 
traces, where seen on edge, as PR, Fig. 62, must cut the projections 
of both surfaces. These two solutions hold good even if the axes 
do not meet but pass each other at right angles. 






















































78 


Engineering Descriptive Geometry 


If tlie axes are not at right angles, modifications must be made, 
and the search for a system of planes making elementary intersec¬ 
tions with both surfaces requires some ingenuity and thought. 

73. Intersection of a Cone and Cylinder: Axes Parallel. —A 
simple case is shown in Fig. 63. Two methods of solution are avail¬ 
able. In one, horizontal planes are used. Each plane, such as T, 



makes circular intersections, with both cone and cylinder, the inter¬ 
sections intersecting at points t and t. A second method is by 
planes perpendicular to H, containing the axis PP'. One plane 
P” is shown by its traces P’R in H and PP" in g, this plane 
being taken so as to give the same point t on the curve and another 
point In the execution of drawings of this class it is natural to 
take the auxiliary planes at regular intervals if the planes are 

parallel to each other, or at equal angles if the planes radiate from 
a central axis. 


























































Intersections of Curved Surfaces 


79 

Problems VII. 

70. An inclined cylinder has one base in H and one in a plane 
parallel to ff. Its axis is P (11, 8, 0), P' (5, 8,16). Its radius is 
4 units. It is intersected by a plane perpendicular to V, its trace 
passing through the points (5,0,0) and (11.0,16). Draw the 
three projections and show one intersecting auxiliary plane by con¬ 
struction lines. 

71. A cone has its vertex in at (6,6,0) and its base in a 
plane parallel to fi, center at (6, 6,12), and radius 5. It is inter¬ 
sected by a plane containing the axis of Y and making angles of 
45° with H and §. Draw the projections. 

72. A cone has its vertex at (2,14,16) and its base is a circle 
in H, center at (8, 8, 0), and radius 6. Find its intersection with 
a vertical plane 4 units from §. 

73. A right circular cylinder has its base in §, center at (0, 8, 8), 
and radius 4. Its axis is 16 units long. Another right cylinder 
has its base in fl-J, center at (8,8,0), radius 5, and axis 16 units 
long. Draw their lines of intersection, the smaller cylinder being 
supposed to pierce the larger. 

74. A right circular cylinder has its base in §, center at (0, 7, 8), 
and radius 4. Its axis is 16 units long. Another right cylinder 
has its base in H, center at (8, 9, 0), radius 5, and axis 16 units 
long. Draw their line of intersection, the smaller cylinder being 
supposed to pierce the larger. 

75. Two inclined circular cylinders of 3 units’ radius have their 
bases in ff and in J-J' (16 units from H). The axis of one is 
P (4,8,0), P' (12,8,16), and of the other is Q (12,8,0), 
Q' (4,8,16). Prove that their line of intersection consists of a 
circle in a plane parallel to H and an ellipse in a plane parallel 
to §. 

76. A sphere has its center at (8, 9, 8), and radius 6 units. A 
vertical right circular cylinder has its top base in ff, center at 
(8, 6, 0), radius 4, and length 16 units. Find the intersections of 
the surfaces. 

77. A right circular cylinder, axis P (0,8,9), P f (16,8,9), 
radius 5, is pierced by a right circular cone. The base of the cone 


80 


Engineering Descriptive Geometry 



is in a plane 16 units from H, center at Q (8, 18, 16), and radius 
6. The vertex of the cone is at Q (8, 8, 0). Find the lines of inter¬ 
section. 

78. An inclined cylinder has an oblique line P (0,11,5), 
P' (16,5,11) for its axis. The radius of the circular base is 4 
units and the planes of the bases are §, and S' parallel to § at 16 
units 7 distance. The cylinder is cut by a plane parallel to V a t 7 
units 7 distance from V- Draw the three projections of the cylinder 
and the line of intersection. 

79. An inclined cylinder has an oblique line P (0,11,5), 
P’ (16,5,11) for its axis. The radius of the circular base is 4 
units, and the planes of the bases are § and S' parallel to § at 16 
units 7 distance. The cylinder is cut by a plane perpendicular to 
V, its trace passing through the points (2,0,0) and (14.0,16). 
Draw the three projections. 


CHAPTER VIII. 


INTERSECTIONS OF CURVED SURFACES; CONTINUED. 

74. Intersection of a Surface of Revolution and an Inclined 
Plane.—In Figs. 64 and 65 a surface of revolution is shown. It is 



cut by an inclined plane perpendicular to IHI in the first case, and 
by one perpendicular to V i n the second case. The planes are 
given by their traces, and the problem is to find the curves of inter¬ 
section. Both solutions make use of cutting planes perpendicular 
to PP', the axis of revolution of the curved surface. 




























































82 


Engineering Descriptive Geometry 


In Fig. 64 a plane T, taken at will perpendicular to PP', cuts 
the surface of revolution in a circular element seen as the straight 
line at' in V- « is projected to J-J and the circle att' drawn. The 
inclined plane whose traces are JK and KL is intersected by the 
plane T in a line whose horizontal projection is the line KL itself. 
t and V (on H) are therefore the intersections of the intersections 
and. are projected to the front elevation, giving points on the re¬ 
quired line of intersection. A system of planes such as T defines 
points enough to fully determine the curve, mtt'n. 

In Fig. 65 the given plane has the traces IX and XZ. The plane 
T intersects the surface of revolution on the circle aid ', and it 



intersects the plane in the line tf, seen on end in V as the point t. 
t and f in ff are points on the required curve of intersection, mtt'n. 

The point of this surface of revolution APC has been given a 
special name. It is an “ ogival point/’ The generating line AP 
is an arc of 60°, center at C, and conversely the generating line PC 
has its center at A. The shell used in ordnance is usually a long 
cylinder with an ogival point. A double ogival surface is produced 
by revolving an arc of 120° about its chord. 

75. Intersection of Two Surfaces of Revolution: Axes Par¬ 
allel.—This problem is illustrated in Fig. 66, where two surfaces of 





















































Intersections of Curved Surfaces 


8:3 


revolution are shown. A horizontal plane T cuts both surfaces in 
circular elements. These elements are drawn in ff as circles abed 
and efgh. t and V are the intersections of the intersections. From 
H t and if are projected to V and §. The problem in Art. 73 is 
but a special case of this general problem. In addition to the Solu¬ 
tion by horizontal planes another solution is there possible, due to 
special properties of the cone and cylinder. 



76. Intersection of Two Surfaces of Revolution: Axes In¬ 
tersecting.—An example of two surfaces of revolution whose axes 
intersect is given by Fig. 67. A surface is formed by the revolution 
of the curve ww' about the vertical axis PP f , and another surface 
by revolving the curve uQ about the horizontal axis QQ'. The in¬ 
tersection of the axes PP f and QQ' is the point p. The peculiarity 







































84 


Engineering Descriptive Geometry 


of this case is that no plane can cut both surfaces in circular ele¬ 
ments. However, a sphere described with the point of intersection 
of the axes as a center, if of proper size, will intersect both surfaces 
in circular elements. V is parallel to both axes and on this pro¬ 
jection a circle is described with p as center representing a sphere. 
The radius is chosen at will. To keep the drawing clear, this 
sphere has not been described on plan or front elevation, as it would 
be quite superfluous in those views. 

The sphere has the peculiarity that it is a surface of revolution, 
using any diameter as an axis. The curve ivw' and the semicircle 
mabn are in the same plane with the axis PP'. When both axes 
are revolved about PP', a and b, their points of intersection, gene¬ 
rate circular elements, which are common to the sphere and to the 
vertical surface of revolution. Therefore, these circles are the in¬ 
tersections of the sphere and the vertical surface. The fi and § 
projections of these circles are next drawn. 

The curve uQ and the semicircle qcdr are in the same plane with 
the axis QQ'. When both axes are revolved about QQ ', their inter¬ 
sections, c and d, generate circles which are common to both sur¬ 
faces, or are their lines of intersection. The circle generated by c 
is drawn in ff and g, hut that generated by d is not needed. 

The three circles aa f , bb ', and cc' appear as straight lines on V? 
but from them the points t and s, the intersections of the intersec¬ 
tions, are determined. These are points on the required curve in 

v. 

The circle aa' appears as a circle ata!t' in J-J, and as a line tt' 
in g. The circle cc' appears as a circle ctc't' in g, and as a line ee' 
in J-J. These circles intersect in f| at t and t', and in g at t and t' 
and s and s'. These are points on the required curves in ff and g. 

For the complete solution, a number of auxiliary spheres, differ¬ 
ing slightly in radius, must be used. 

77. Intersection of a Cone and a Non-Circular Cylinder. —A 
non-circular cylinder is a surface created by a line which moves 
always parallel to itself, being guided by a curve lying in a plane 
perpendicular to the generating line. This curve, called the direc¬ 
trix. is usually a closed curve. The cross-section of such a cylinder 
is everywhere similar to the directrix. 


Intersections of Curved Surfaces 


85 


This fact may be utilized to advantage in some cases. In Fig. 
68, an oblique cone and a non-circular cylinder intersect. The 
directrix of the cylinder is a pointed oval curve, abed in ff. Hori¬ 
zontal planes, as T'T, intersect the cylinder in a curve identical in 
shape with its directrix, so that its projection on fl coincides with 
the projection of the directrix on ff. The intersection with the 
cone is a circle, mt'tn, and the intersections of the intersections are 
the points t. 



78. Alteration of a Curve of Intersection by a Fillet. —In Fig. 

69 a hollow cone and a non-circular cylinder, abed in ff, intersect. 
On the left half the unmodified curve of intersection is traced by 
the method of the preceding article, no construction lines being 
shown however, as the case is very simple. On the right half the 
curve is modified by a fillet or small arc of a circle which fills in 
the angular groove. The fillet whose center is at g modifies that 
point of the line of intersection marked c. The top of the circular 
arc marks the point where an ff or § projector is tangent to the 
surface. 


7 




































86 


Engineering Descriptive Geometry 


The corresponding crest to the fillet at other positions on the 
curve of intersection is traced as follows: If a line drawn through 
k and parallel to PG, the generating line of the cone, is used as 
a new generator it will by its rotation about PP' create a new 
cone, on the surface of which the required line of the crests of the 



fillets must lie. If a line mn, parallel to cc', the generating line of 
the cylinder, is moved parallel to cc', and at a constant distance 
from the surface of the non-circular cylinder, it will generate a 
new non-circular cylinder on the surface of which the required 
path of the point k must lie. The directrix of this new cylinder is 
drawn in fi, the line rms, as shown. The intersection of these two 

























Intersections of Curved Surfaces 


87 



Fig. 70. 
















































































88 


Engineering Descriptive Geometry 


new surfaces, found by the method used above (or by planes per¬ 
pendicular to fl-fl through the axis PP'), is the required path of ft 
or the line which appears on V and S- The line rms, representing 
the same path on is not properly a line of the drawing and is 
not inked except as a construction line. 

79. Intersection of a Helicoidal Surface and a Plane. —In Fig. 
70 there is shown a long-pitched screw having a triple thread, such 
as is often employed for a “ worm.” To the left is shown a partial 
longitudinal section giving the generating lines. In V the con¬ 
cealed parts of the helical edges are omitted, except in the cases of 
one of the smaller and one of the larger edges. The plane whose 
trace on V is PL is perpendicular to the axis, and terminates the 
screw threads. The intersection of this plane with the screw 
threads is the curve of intersection to be drawn on f|. It is deter¬ 
mined by passing planes containing the axis of the worm. One of 
these is shown by its traces PR and RR r . 

From points a and b in the plan corresponding points are plotted 
on the front elevation, a falling on the helix of small diameter 
(extended in this case), and b on the helix of large diameter. This 
element ab of the helix is seen to pierce the plane KL at ft. This 
point ft is projected to the plan and is one of the points on the 
required curve mien. 


Problems VIII. 

(For units, use inches on blackboard or wire-mesh cage, or small 

squares on cross-section paper.) 

80. An anchor-ring is formed by revolving a circle of 6 units’ 
diameter about a vertical axis, so that its center moves in a circle 
of 10 units’ diameter, center at Q (8,8,8). The anchor-ring is 
intersected by a plane parallel to V through the point A (8, 6, 8) 
and by another plane parallel to V through the point B (8, 4, 8). 
Draw the projections of the ring, the traces of the planes and the 
lines of intersection. 

81. The same anchor-ring is intersected by a plane perpendicu¬ 
lar to V, having a trace passing through the points C (0, 0, 2) and 
D (8, 0, 8). Make the descriptive drawing and show the true shape 
of the lines of intersection. 


Intersections of Curved Surfaces 


89 


82. The same anchor-ring is intersected by a right circular 
cylinder, axis P (12, 8, 0 ), P' (12, 8,16), and diameter of 4 units. 
Make the descriptive drawing of the anchor-ring’, imagining it to 
be pierced by the cylinder. 

83. An anchor-ring has an axis P (0,8,8), P' (16,8,8). Its 
center moves in a plane 10 units from § describing a circle of 8 
units’ diameter. The radius of the describing circle is 3 units. It 
is intersected by an ogival point whose axis is a vertical line 
Q (7, 8, 3f), Q' (7, 8,16). The generating line of the ogival point 
is an arc of 60°, with center at (0, 8, 16), and radius 14 units, so 
that the point Q is the vertex and point Q' is the center of the circu¬ 
lar base of 7 units’ radius. The axes intersect at P (7, 8, 8). Draw 
the projections and the line of intersection, front and side eleva¬ 
tions only. 

84. The line P (4,13, 8), P' (16, 8, 8) is the chord of an arc of 
45°, whose radius is 13 units. The arc is the generating line of a 
surface of revolution of which PP' is the axis. Draw the projection 
on J-J. Draw the end view on an auxiliary plane JJ perpendicular 
to PP', the trace of U on Jf-fl intersecting OX at (16, 0, 0). The 
surface is intersected by a plane perpendicular to ff and contain¬ 
ing the line PP'. Draw the line of intersection in V- 

85. The same surface is intersected by a plane perpendicular to 
Iff whose trace in ff passes through the points (4, 10, 0) and 
(16, 5, 0). Draw the line of intersection on V- 

86. The line P (3, 8, 8), P' (13, 8, 8) is the chord of an arc of 
60°, radius 10 units. It is the axis of revolution of a surface of 
which the arc is the generating line. It is intersected by a right 
circular cone having its vertex at Q (8, 8, 2), and center of base at 
Q' (8, 8,12), radius of base 6 units. Draw the line of intersection. 

87. A non-circular cylinder lias its straight elements, length 16 
units, perpendicular to ff, passing through the points of a smooth 
curve through the points A (14,6,0), B (12,4,0), C (10,4,0), 
D (8, 5, 0), E (5, 8, 0), F (2,13, 0). It is pierced by a cylinder 
whose base is in Y? whose axis is perpendicular to V at the point 
(8,0,8), and whose radius is 5 units and length 14 units. Find 
the line of intersection in S. 


90 


Engineering Descriptive Geometry 


\ 


88. The line P (8,8,2), P (8, 8, 14) is the axis of a right cir¬ 
cular cylinder of 6" diameter. Projecting from the cylinder is 
an helicoidal surface, of 12 units’ pitch, of which G (5, 8, 2), 
G' (1,8,2) is the generating line. The helicoid is intersected 
by a plane perpendicular to ]fi whose trace in ff passes through the 
points (5,0,0) and (16,11,0). Draw the plan and front eleva¬ 
tion of the cylinder and helicoid and plot the line of intersection 
with the plane. 

89. The helicoidal surface of Problem 87 is intersected by a right 
circular cylinder whose axis Q (12, 8, 2), Q' (12, 8, 14) is parallel 
to PP'. The radius of the cylinder is 3 units. Draw the line of 
intersection. 


CHAPTER IX. 

DEVELOPMENT OF CURVED SURFACES. 


80. Meaning of Development as Applied to Curved Surfaces.— 

Many curved surfaces may be developed on a plane in a manner 
similar to the development of prisms and pyramids explained in 
Articles 45 and 46. By development, is meant flattening out, 
without stretching or otherwise distorting the surface. If a curved 
surface is developed on a plane and this portion of the plane, called 
“ the development of the surface,” is cut out, this development may 



be bent into the shape of the surface itself. The importance of 
the process comes from the fact that many articles of sheet metal 
are so made. If a sheet of paper is bent in the hands to any fan¬ 
tastic shape, it will always be found that through every point of 
the paper a straight line may be drawn on the surface in some one 
direction, the greatest curvature of the surface at this point being 
in a direction at right angles to this straight line element through 
the point. The surfaces which can be formed by twisting a plane 
surface without distortion are called surfaces of single curvature. 
The curved surfaces, therefore, which are capable of development 
are only those which are surfaces of single curvature and have 
straight line elements, but not by any means all of these. All forms 




92 


Engineering Descriptive Geometry 


of cylinders and cones, right circular, oblique circular, or non¬ 
circular, may be developed. The helicoidal surfaces, illustrated by 
Figs. 55 and 56, though having straight elements, cannot be de¬ 
veloped, nor can the hyperboloid of revolution, a surface generated 
by revolving a straight line about a line not parallel nor intersect¬ 
ing. Figs. 71 and 72 are perspective drawings showing the process 
of rolling out or developing a right circular cylinder and a right 
circular cone. 

81. Rectification of the Arc of a Circle.—In developing curved 
surfaces it frequently happens that the whole or part of the cir¬ 
cumference of a circle is rolled out into a straight line. Since the 
surface must not be stretched or compressed, the straight line must 
be equal in length to the arc of the circle. This process of finding 
a straight line equal to a given arc is called rectifying the arc. No 



G 


Pig. 73. 


Fig. 74. 


absolutely exact method is possible, but methods are known which 
are so nearly exact as to lead to no appreciable error. These have 
the same practical value as if geometrically perfect. 

In Fig. 73, AB is the arc of a circle, center at C. For accurate 
work the arc should not exceed 60°. It is required to find a 
straight line equal to the given arc. Draw AH, the tangent at one 
extremity, and draw AB, the chord. Bisect AB at D. Produce the 
chord and set off AE equal to AD. With E as a center, and with 
EB as a radius, describe the arc BF, meeting A H at F. Then 
AF -=arc AB, within one-tenth of one per cent. 

In this figure, and in the two following ones, the arc and the 
straight line equal to it are made extra heavy for emphasis. 
















Development of Curved Surfaces 


93 


82. Rectifying a Semicircle. —A second method, applicable par¬ 
ticularly to a semicircle, was recently devised by Mr. George Pierce. 
In Fig. 74 the semicircle AFB is to be rectified. A tangent BC, 
equal in length to the radius, is drawn at one extremity. Join AC, 
cutting the circumference at D. Lay off DE-DC, and join BE, 
producing BE to the circumference at F. Join AF. Then the 
triangle AEF, shown lightly shaded, has its periphery equal to the 
semicircle AFB, within one twenty-thousandth part. The peri¬ 
phery may be conveniently spread into one line by using A and E 
as centers, and with AF and EF as radii, swinging F to the left to 
C and to the right to H on the line AF extended. GH is the recti¬ 
fied length of the semicircle. 

83. To Lay Off an Arc Equal to a Given Straight Line. —This 
inverse problem, namely to lay off on a given circle an arc equal to 



a given straight line,, frequently arises. In Fig. 75 a line AB is 
given. It is required to find an arc of a given radius AC equal to 
the given line AB. At A erect a perpendicular, making AC equal 
to the given radius,'and with C as a center describe the -arc AF. 
On AB, take the point D at one-fourth of the total distance from 
A. With D as center and DB as a radius, draw the arc BF, meet¬ 
ing AF at F. AF is the required arc, equal to AB. 

This process is also accurate to one-tenth of one per cent if the 
arc AF is not greater than 60°. If in the application of this process 
to a particular case the arc AF is found to be greater than 60°, the 
line AB should be divided into halves, thirds or quarters, and the 
operation applied to the part instead of to the whole line. 














































































94 


Engineering Descriptive Geometry 


I 


84. Development of a Straight Circular Cylinder.—In Fig. 60 

let the intersecting cylinders represent a large sheet-iron ventilat¬ 
ing pipe, with two smaller pipes entering it from either side. Such 
a piece is called by pipe fitters a “ cross.' 7 The problem is to find 
the shape of a flat sheet of metal which, when rolled up into a 
cylinder, will form the surface of the vertical pipe, with the open¬ 
ings already cut for the entrance of the smaller pipes. Before 
developing the large cylinder, it must be considered as cut on the 
straight element BB'. After the pipe is formed from the develop¬ 
ment used as a pattern, the element BE' will be the location of a 
longitudinal seam. 

A rectangle, Fig. 76, is first drawn, the height BB' being equal 
to the height of the cylinder and the horizontal length being equal 
to the circumference of the base BCD A. (This length may be best 
found by Mr. Pierce's method, which gives the half-length, BD.) 
On the drawing, Fig. 60, the base BCDA must be divided into 
equal parts, 24 parts being usually taken, as they correspond to 
arcs of 15°, which are easily and accurately constructed with the 
draftsman’s triangles. Only 6 of these 24 parts are required to be 
actuallv marked on Fig. 60, as the figure is doublv svmmetrical 
and each quadrant is similar to the others. On Fig. 76 the line 
BCDAB is divided into 24 parts also, the numbering of the lines 
of division running from 0 to 6 and back to 0 for each half-length 
of the development. In V of Fig. 60, draw the elements corre¬ 
sponding to the points of division. The elemnt IV already drawn 
corresponds to Xo. 4, and BB' and CC' correspond to Xos. 0 and 6. 
The others are not drawn in Fig. 60, to avoid complicating the 
figure, but would have to be drawn in practice before constructing 
the development. On the four elements which are numbered 4 on 
the development, Fig. 76, lay off the distances Jr equal to Jr in 
Fig. 60. On the two elements, Fig. 76, numbered 6, lay off Cc or 
Aa equal to Cc of Fig. 60, and imagine the proper distances to be 
laid off on elements numbered 3 and 5. Smooth curves through 
the points thus plotted are the ovals which must he cut out of the 
sheet of metal to give the proper-shaped openings for the small 
pipes. 

When it is known in advance that the surface of such a cylinder 


Development of Curved Surfaces 


95 


as that in Fig. 60 must be developed, it is often possible to so 
choose the system of auxiliary intersecting planes used to define 
the curve of intersection as to give the required equally spaced 
straight elements for the development. 

The smaller cylinder may be developed in the same way. A new 
system of equally spaced straight elements would probably have to 
be chosen for this cylinder. 

85. Development of a Right Circular Cone. —The cone of Fig. • 
63 has been selected for this illustration. Imagine it to be cut on 
the element PB and flattened into a plane. The surface takes the 



form of a sector of a circle, the radius of the sector being the slant 
height of the cone (or length of the straight element), and the arc 
of the sector being equal in length to the circumference of the base 
of the cone. Several means of finding the length of the arc of the 
sector are available. 

The most natural method is to rectify the circumference of the 
base and then, with the slant height as radius, to draw an arc and 
to lay out on the arc a length equal to this rectified circumference. 
In Fig. 63 suppose that the semi-circumference ABC (in ff) has 
been rectified by Pierce’s method. In Fig. 77 let an arc be drawn 
with radius PB equal to PB in g, Fig. 63, and from B draw a 
tangent BE equal to one-half the rectified length of the semi-cir¬ 
cumference. Find the arc BC equal to BE by the method of Art. 









96 


Engineering Descriptive G-eometry 


l 


83, Eig. 75. BC is one-fourth of the required arc, and corresponds 
to the quadrant BC in H, Eig. 63. Divide the arc BC and the 
quadrant BC into the same number of equal parts, numbering 
them from 0 to 6, if 6 parts are chosen. Repeat the divisions in 
the arc CD (equal to BC), numbering the points of division from 
6 down to 0, this duplication of numbers being due to the symmetry 
of the H projection of Fig. 63, about the line APC. In Fig. 63, 
• as in Fig. 77, the points 0 to 6 are all supposed to be joined to P, 
the only straight elements actually shown there being P0, P4, and 
P6. 

On the elements P4 of the development lay off the true length 
of the line Pt (and the true length of the line Pt' also). Pt is an 
oblique line, but if its ff projector-plane (Pt in ff, Fig. 63) be 
revolved up to the position Pm, the point t in V moves to m, and 
Pm is the true length of Pt. The distance Pg (V , in Fig. 63) is 
laid off on P6 of the development. 

When the proper distances have been laid off on the elements 
P2, P 3 and P5, a smooth curve may be drawn through the points. 
The sector, with this opening cut in it, is the pattern for forming 
the cone out of sheet iron or any thin material. 

If the ratio of PA to P'A in V, Fig. 63, can be exactly deter¬ 
mined, the most accurate method of getting the angle of the sector is 
by calculation, for the degrees of arc in the development are to the 
degrees in the base of the cone (360°) as the radius of the base of 
the cone is to the slant height. In this case P'A is f P.4. The 
sector in Fig. 75 subtends f x360°, or 216°. In the use of this 
method a good protractor is required to lay out the arc. 

Problems IX. 

90. Draw an arc of 60° with 10 units’ radius. At one end draw 
a tangent and on the tangent lay off a length equal to the given 
line. On the tangent lay off a length of 8 units, and find the length 
of arc equal to this distance. 

91. An arc of 12 units’ radius, one of 9 units’ radius, and a 
straight line are all tangent at the same point. Find on the tan¬ 
gent the straight line equal in length to 45° of the large arc. Find 
the length on the other arc equal to this length on the tangent and 
show that it is an arc of 60°. 


Development of Curved Surfaces 


97 


92. Rectify a semicircle of 10 units’ radius and compare the 
length with the calculated lengthy 31.4 units. 

93. A rectangle 31.4 units by 12 units is the developed area of a 
cylinder of 10 units’ diameter. A diagonal line is drawn on the 
development, which is then rolled into cylindrical form. Plot the 
form taken by the diagonal and show that it is a helix of 12 units’ 
pitch. 

94. A right circular cone has a base of 10 units’ diameter, and 
a vertical height of 12 units. Its slant height is 13 units. Calcu¬ 
late the angle of the sector which is the developed surface of the 
cone. Find this angle by rectifying the circumference of the cone, 
and by finding the arc equal to the rectified length. (This last 
operation must be performed on one-third or one-quarter of the 
rectified length, to keep the accuracy within one-tenth of one per 
cent.) 

95. A semicircle, radius 10 units, is rolled up into a cone. What 
is the radius of the base? What is the slant height? What is the 
relation between the area of the curved surface of the cone and the 
area of the base ? 

96. A right circular cylinder, such as Fig. 49, is of 6.367 units’ 
diameter, and 12 units’ height. It is intersected by a plane per¬ 
pendicular to V through the points G and A'. Draw plan, front 
elevation and the development of the surface. 

97. A right circular cone, like that of Fig. 51, has its front ele¬ 
vation an equilateral triangle, each side being 10 units in length. 
From A v a perpendicular is drawn to P V C V cutting it at E. If this 
line represents a plane perpendicular to V? draw the development 
of the cone with the line of intersection of the cone and plane traced 
on the development. 

98. A right circular cylinder, standing in a vertical position, as 
in Fig. 49, diameter 7 units, and length 10 units, is pierced from 
side to side by a square hole 3-J units on each edge, the axis of the 
hole and the axis of the cylinder bisecting each other at right 
angles. Draw the development of the surface. 

99. A sheet of metal 22 units square with a hole 11 units square 
cut out of its middle, the sides of the hole being parallel to the 
edges of the sheet, is rolled up into a cylinder. Draw the plan, 
front and side elevations of the cylinder. 




CHAPTER X. 


STRAIGHT LINES OF UNLIMITED LENGTH AND THEIR 

TRACES. 


86. Negative Coordinates.—We have dealt only with points hav¬ 
ing positive or zero coordinates, and the lines and planes have been 



iWi 



Fig. 78. 




limited in their extent, or, if infinite, have extended indefinitely 
only in the positive directions. As it becomes necessary at times 
to trace lines and planes in their course, no matter if they cross 
the reference planes into new regions of space, the use and meaning 
of negative coordinates must be explained. The value of the x 
coordinate of a point is the length of the § projector or perpen¬ 
dicular distance from the point to the side reference plane g. (See 
Figs. 6 and 7, Art. 9.) If this value decreases gradually to zero, 










Lines or Unlimited Length: Their Traces 


99 


the point moves towards § until it lies in g itself. If this value 
becomes negative, it is clear that the point crosses the side reference 
plane into a space to the right of it. 

For example, a point P, having a variable x coordinate, but hav¬ 
ing its y coordinate always equal to 4 and its z coordinate equal to 
2, is a point moving on a line parallel to the axis of X. If x de¬ 
creases to zero, it is on g at the point marked P s in Fig. 78. If 
the x coordinate decreases further, reaching a value of —3, it 
moves to the point P in that figure. Fig. 78 is the perspective 
drawing of a point P ( — 3,4,2). The y and z projectors cannot 
project the point P to H and V in their customary positions, but 



project it upon parts of those planes extended beyond the axes of 
Y and Z, as shown. In Fig. 79, the corresponding descriptive 
drawing, it must be understood that the plane H, extended, has 
been revolved with H, about the axis of X, into the plane of the 
paper, V, and g has been revolved as usual about the axis of Z , 
coming into coincidence with V, extended. This “ development ” 
of the planes of reference is exactly as described in Art. 7. It is 
noticeable that the x coordinate of P is laid off to the right of the 
origin instead of to the left. Ph lies, therefore, in the quadrant 
which usually represents no plane of projection, and P v lies in the 
quadrant which usually represents g. P s lies in its customary 
place, since both y and z. the coordinates which alone appear in g, 
are positive. 














100 


Engineering Descriptive Geometry 


It is evident that the laws of projection for fj[, V and S> Art. 
11, have not been altered, but simply extended. P h and P v are in 
the same vertical line; P v and P s are in the same horizontal line; 
and the construction which connects Ph and P s still holds good. 

In Fig. 79 the space marked § represents not only § but V 
extended as well. 

In Fig. 80 is represented a point P (3, —2, 3), having a negative 
y coordinate. The point is in front of V? at 2 units' distance, not 
behind V- The projection on H, instead of being above the axis 
of X a distancee of 2 units, is below it by the same amount. So also 
the projection on § is to the left of the axis of Z, a distance of 2 
units, instead of the the right of it. After developing the reference 
planes in the manner of Art. 7, plane H, extended, has come into 
coincidence with V> and plane §, extended, has also come into co¬ 
incidence with V. Thus the field representing V represents also 
the other two reference planes, extended. 

In Fig. 81 a point P (2, 2, —3) having a negative z coordinate 
is represented. The point is above fff 3 units, instead of below H, 
at the same perpendicular distance. P projects upon V on V 
extended above the axis of X. After developing the reference 
planes, plane fj comes into coincidence with V extended. P s is 
on § extended above the axis of Y, and therefore after develop¬ 
ment it occupies the so-called “ construction space." 

Points having two or three negative coordinates may be dealt 
with in the same manner, but are little likely to arise in practice. 

It is evident that subscripts must be used invariably, to prevent 
confusion whenever negative values are encountered. 

87. Graphical Connection Between P and P s . —In Figs. 79, 80 
and 81, P h and P s are connected by a construction line PhfhfsPs in 
a manner which is an extension of that shown by Fig. 7, Art. 9. 
Note that the quadrant of a circle connecting P h and P s must be 
described always on the construction space or on the field devoted 
to V? never on the fields devoted to ff or §. 

88. Traces of a Line of Unlimited Length, Parallel to an Axis.— 
A straight line which has no limit to its length, but extends in¬ 
definitely in either direction, must necessarily have some points 
whose coordinates are negative. In passing from positive to nega- 



Lines of Unlimited Length: Their Traces 


101 


tive regions the line must pass through some plane of reference 
(having one of its coordinates zero at that point), and the point 
where it pierces a plane of reference is called the trace of the line 
on that plane of reference, the word trace being used to indicate a 
“ track ” or print showing the passage of the line. 

Lines parallel to the axes have been used freely already. An J-J 
projector is simply a vertical line or line parallel to the axis of Z. 
Any perspective figure showing a point P and its horizontal pro¬ 
jection P h will serve as an illustration of this line, as PP h in Fig. 
6, Art. 9. 



Imagine PPh to be extended in both directions as an unlimited 
straight line. Then P h is the trace of the line on H. In Fig. 7, 
the point Pu itself is the ff projection of the line. P v e, extended 
in both directions, is the vertical projection and P s f s is the side 
projection. Thus it is seen that a vertical line has but one trace, 
that on the plane to which it is perpendicular. PP V may be taken 
as an illustration of a line parallel to the axis of Y, and PP S of one 
parallel to the axis of X. A better example of this latter case is 
shown in Figs. 15 and 16, Art. 16. The line BAA S , perpendicular 
to §, has its trace on § at A s . 

8 















102 


Engineering Descriptive Geometry 


89. Traces of an Inclined Straight Line.—An inclined line such 
as AB in Figs. 82 and 83 pierces two reference planes as at A and 
B, but as it is parallel to the third reference plane, §, it has no 
trace on §. The peculiarity of the descriptive drawing of this line, 
Fig. 83, is the apparent coincidence of the ff and V projections 
as one vertical line. The § projection is required to determine the 
traces A and B. 

90. Traces of an Oblique Straight Line: The H and V Traces.— 

An oblique line, if unlimited in length, must pierce each of the 
reference planes, since it is oblique to all three. Any line is com¬ 



pletely defined when two points on the line are given. If two 
traces of a straight line are given, the third trace cannot be assumed, 
but must be constructed from the given conditions b}r geometrical 
process. It will always be found that of the three traces of an 
oblique line one trace at least has some negative coordinate. 

As the complete relation between the three traces is somewhat 
complicated, the relation between two traces, as, for instance, ff 
and V traces, must be considered first. Two cases are shown, the 
first by Figs. 84 and 85, and the second by Figs. 86 and 87. The 
line AB is the line whose traces are A (5,0,4) and B (2,4,0). 
The line CD is the line whose traces are C (7, 0, 5) and D (2, 4, 0). 











Lines of Unlimited Length: Their Traces 


103 


From the descriptive drawing of AB, Fig. 85, it is seen that the 
H projection of the line cuts the axis of X vertically above the 
trace on V, and that the V projection cuts the axis of X vertically 
under the trace on ff. It may be noted that the two right triangles 
AjtBBv and B v AA n have the line A h B v on the axis of X as their 
common base. From the descriptive drawing of the line CD, Fig. 
87, it is seen that the effect of the vertical trace C having a nega¬ 
tive z coordinate simply puts C (on V) above Ch, instead of below it. 
The two right triangles Ct,DD v and D v CCn have the line Ci,D v on 
the axis of X, as their common base, but the latter triangle is above 
the axis instead of in its normal position. 



91. Traces of an Oblique Straight Line: The V and S Traces.— 

Figs. 88 and 89 show two lines piercing V and §. 

The line AB pierces V at A and § at B. The two right triangles 
A S AB V and B r BA s have their common base A S B V on the axis of Z. 

The line CD pierces V at C and § extended at D, the point D 
having a negative y coordinate. The right triangles C S CD V and 
D v DCs have their base D V C S in common on the axis of Z, but in the 
descriptive drawing D V DC S lies to the left of the axis of Z instead 
of to the right, owing to the point D having a negative y coordinate. 

















104 


Engineering Descriptive Geometry 


I 


92. Traces of an Oblique Straight Line: The ff and S Traces.— 

Figs. 90 and 91 show two lines piercing and §. 

The line AB pierces H at A and § at B. The triangles A s ABh 
and BhBA s have their common base A s Bn on the axis of Y, Fig. 90, 
but in the descriptive drawing the duplication of the axis of Y 
causes this base A s B h to separate into two separate bases, one on 
OY h and one on OY s . Otherwise, there has been no change. 

The line CD pierces H at (7 and § extended at D, the point D 
having a negative 2 coordinate. In Fig. 90 C s CD h and D),DC S have 



their common base C s Du on the axis of Y, but in the descriptive 
drawing C s Dh appears in two places. The triangle D] t DC s lies above 
§ in the “ construction space,” or on § extended, since D has a 
negative z coordinate. 

93. Three Traces of an Oblique Straight Line.—Figs. 92 and 93 
show an oblique straight line ABC piercing V at A, H at B, and 
§ extended at C. Since the line is straight, the three projections 
of the line AB V C V , A S B S C and Ai,BCh are all straight lines. In the 
perspective drawing, Fig. 92, part of the V projection is on V ex- 
tended and part of the § projection on § extended. 














Lines of Unlimited Length: Their Traces 


105 


In the descriptive drawing, Fig. 93, the relation between A and 
B is the same as that in Fig. 85, as shown by the two triangles 
AhAB v and B v BAn, or the quadrilateral A h AB v B. The relation 
between A and C, as shown by the quadrilateral A S AC V C, is the 
same as that between A and B, Fig. 89, as shown by the quadri¬ 
lateral A S AB V B. The relation between B and ( 7 , Fig. 93, as shown 
by the two triangles B s BC h and CnCB s , is the same as that between 
C and D of Fig. 91, as shown by the triangles C a CD h and D h DC 
Uo new feature has been introduced. 



94. Paper Box Diagram.—To assist in understanding Figs. 92 
and 93, a model in space should be made and studied from all 
sides. The complete relation of the traces is then quickly grasped. 
Construct the descriptive drawing, Fig. 93, on coordinate paper, 
using, as coordinates for A, B and C, (15,0,12), (5,12,0), and 
(0, 18, —6). Fold into a paper box after the manner of Fig. 9, 
Art. 12, having first cut the paper on some such line as mn, so that 
the part of the paper on which G is plotted may remain upright, 
serving as an extension to §. It will be found that a straight wire 
or long needle or a thread may be run through the points A, B and 
C, thus producing a model of the line and all its projections. 
























106 


Engineering Descriptive Geometry 


f 


95. Intersecting Lines. —If two lines intersect, their point of 
intersection, when projected upon any plane of reference, must 
necessarily be the point of intersection of the projections on that 
plane. For example, a line AB intersects a line (D at E. Project 
E upon a plane of reference, as JHI - Then Eh must be the point of 
intersection of A%Bh and CiAhi. In the same way E v must be the 
point of intersection of A V B V and C V D V , and E s of A S B S and C 8 D S . 

To determine whether two lines given by their projections meet 
in space or pass without meetings the projections on at least two 
reference planes must be extended (if necessary) till they meet. 
Then for the lines themselves to intersect, the points of intersec¬ 
tion of the two pairs of projections must obey the rules of pro¬ 
jection of a point in space (Art. 11). Thus if AhBn and ChDj, are 
given and meet at a point vertically above the point of intersection 
of A V B V and C V D V , the two lines really meet at a point whose pro¬ 
jections are the intersections of the given projections. If this con¬ 
dition is not filled the lines pass without meeting, the intersecting 
of the projections being deceptive. 

96. Parallel Lines. —If two lines are parallel, the projections of 
the lines on a reference plane are also parallel (or coincident). 
For, the two lines make the same angle with the plane of pro¬ 
jection; their projector-planes are parallel; and the projections 
themselves are parallel. 

Thus if a line AB is parallel to another line CD. then Ai,Bi, must 
be parallel to (7/,A,, A V B V to C V D V , and A S B S to C S D S . If the two 
lines lie in a plane perpendicular to a plane of projection—for 
example, perpendicular to —then the ffl projector-planes coin¬ 
cide and the H projections also coincide. The V and § projections 
are parallel but not coincident. 

If two lines do not fill the conditions of intersecting or of parallel 
lines, they must necessarily be lines which pass at an angle without 
meeting. 


Lines of Unlimited Length: Their Traces 


107 


Problems X. 

100. Plot the points A (8, 6, —4 ),B (7, -3, 5), C (-7,0,12). 

101. Plot the points A (6, —10, 3), B (0, 0, — 5), O ( — 6,5,4). 

102. Make a descriptive drawing of a line 26 units long through 
the point P (—8,4,9), perpendicular to g. What traces does it 
have ? What are the coordinates of its middle point ? 

103. A line is drawn from P (12, 5, 16) perpendicular to ff. 
Make the descriptive drawing of the line, and of a line perpen¬ 
dicular to it, drawn from Q (0, 0, 8). What is the length of this 
perpendicular line, and where are its traces? 

104. A straight line extends from A (8,12,0) through D 
(8, 6, 8) for a distance of 20 units. Make the descriptive drawing 
of the line. Where are its traces and its middle point ? 

105. A straight line pierces H at A (8, 6,0) and V a t B 
(8, 0, 12). Draw its projections. Where is its trace on g? What 
are the coordinates of D, its middle point? 

106. A straight line extends from P (15,6,16) through A 
(3, 6, 0) to meet g. Make the descriptive drawing and mark the 
traces on H and g. 

107. Draw the lines A (16, 11, 8), B (4, 8, 2); O (12, 5, 10), 
D (0,2,4); and E (11,3,0), F (5,15,16). Which pair meet, 
which are parallel, and which pass at an angle? What are the 
coordinates of the point of intersection of the pair which meet? 

108. The points A (8, 0, 12), B (0, 8, 6) and C ( — 8, 16, 0) are 
the traces of a straight line. Make the descriptive drawing of the 
line. 

109. The points A (8, —4,0), D (4,4,6) and E (2,8,9) are 
on a straight line. Find the trace B where it pierces V and the 
trace C where it pierces g. 



CHAPTER XI. 

PLANES OE UNLIMITED EXTENT: THEIR TRACES. 


97. Traces of Horizontal and Vertical Planes. —The lines of 

intersection of a plane with the reference planes are called its 
traces. Planes of unlimited extent may be of three kinds, parallel 
to a reference plane, inclined, or oblique. Unlimited planes of the 
first two classes have been dealt with already, but for the sake of 
precision may be treated here again to advantage. 

A horizontal plane is one parallel to H, and the trace of such a 
plane on V is a line parallel to the axis of X , and the trace on § 
is a line parallel to the axis of Y. These traces meet the axis of Z 
at the same point and appear on the descriptive drawing as one 
continuous line. There is of course no trace on f|. In Fig. 58, 
Art. 67, the plane T, represented by its traces T'T on V and TT" 
on §, is a horizontal plane. These traces are not only the intersec¬ 
tions of T with ff and §, but T is “ seen on edge" in those views. 
Every point of the plane T, when projected upon lies somewhere 
on the line T'T, extended indefinitely in either direction. 

A vertical plane parallel to V has for its traces a line on 
parallel to the axis of X, and on 5 a line parallel to the axis of 
Z, with no trace on V- These traces meet the axis of Y at the 
same point, and appear on the descriptive drawing as two lines at 
right angles to this axis, the point on Y separating into two points 
as usual. In Fig. 57, Art. 66, a vertical plane R, parallel to V? is 
represented by its traces R'R on ff and RE" on §. 

A vertical plane parallel to § has for its trace on ff a line paral¬ 
lel to the axis of Y, and for its trace on V a line parallel to the 
axis of Z, with no trace on §. These traces meet the axis of X at 
the same point and appear on the descriptive drawing as one con¬ 
tinuous line. 

98. Traces of Inclined Planes. —Inclined planes are those per¬ 
pendicular to one reference plane, hut not to two reference planes. 
The auxiliary planes of projection have been of this kind. In 


Planes of Unlimited Extent: Their Traces 


109 


Fig. 20, Art. 22, the plane HJ, perpendicular to JHJ, has the line 
MX for its trace on HI, and XN for its trace on V. In the de¬ 
scriptive drawing, Fig. 21, MX and XN V are these traces. 

If in Fig. 20 both HJ and § are imagined to be extended towards 
the eye, they will, intersect in a line parallel to OZ. This § trace 
will be on § extended, and every point of it will have the same 
negative y coordinate. Of the three traces of U, two are vertical 
lines, and one only, MX, is an inclined line. The plane in Fig. 64, 
Art. 74, may be taken as a second example of an inclined plane 
perpendicular to Hi* The trace on § is not a negative line in this 
case, but is a vertical line on § to the right of the axis of Z at a 
distance equal to OJ. 

In Fig. 57, Art. 66, IJ, JK and EL are the three traces of an 
inclined plane perpendicular to V- In every case of an inclined 
plane the inclined trace is on that reference plane to which it is 
perpendicular, and shows the angles of the inclined plane with one 
or both of the other reference planes. 




99. Traces of an Oblique Plane: All Traces “Positive.” —The 

general case of an oblique plane is shown in Fig. 94. The plane 
P is represented as cutting the cube of reference planes in the lines 
marked PJI, PV and PS. These lines are the traces of the plane 
P, and may be understood to extend indefinitely, the plane itself 
extending in all directions without limit. They are shown limited 
in Fig. 94 in order to make a more realistic appearance. PH, P T 
and PS are used to define the three traces. 









110 Engineering Descriptive Geometry 

Where PIP and PV meet we have a point common to three planes, 
P, H and V- Since it is common to H and V it is on the line of 
intersection of IHI and V., or i n other words it is on the axis of X. 
This point is marked a. In the same way PH and PS meet at b 
on the axis of Y, and PV and PS meet at c on the axis of Z. 

The descriptive drawing, Fig. 95, is obvious from the explana¬ 
tion of the perspective drawing. From Fig. 95 it is evident that 
if two traces of a plane are given the third trace can be determined 



by geometrical construction. Thus, if PH and PV are given, PS 
may be defined by extending PIP to b on the axis of Y and extend¬ 
ing PV to c on the axis of Z. The line joining be is the required 
trace of the plane on §. If any two points on one trace are given, 
and any one point on a second trace, the whole figure may he com¬ 
pleted. Thus any two points on PPI define that line and enable a 
and b to be found. A third point on PV, taken in conjunction 
with a, defines PV, and enables c to be located, be, as before, 
defines the trace PS. This is an application of the general prin¬ 
ciple that three points determine a plane. 








Planes of Unlimited Extent: Their Traces 


111 


100. Traces of an Oblique Plane: One Trace “ Negative.” —In 

Figs. 94 and 95 the plane P lias been so selected that all traces have 
positive positions. These are the portions usually drawn. Of 
course each trace may be extended in either direction, points on 
the trace then having one or more negative coordinates. Any 
trace having points all of whose coordinates are positive, or zero, 
may be called a positive trace. 

In Fig. 96 a plane P is shown, intersecting J-fl and V in the. 
“ positive ” traces PH and PV. The third trace, PS, in this case, 
has no point all of whose coordinates are positive. In the descrip¬ 
tive drawing, Fig. 97, the two positive traces, meeting at a on the 



axis of X, are usually considered as fully representing the plane P. 
From these lines PH and PV, alone, the imagination is relied upon 
to “ see the plane P in space,” as shown by Fig. 96. 

In Fig. 98, the plane Q is represented. Ordinarily the positive 
traces QV and QS, meeting at c on the axis of Z, are the only 
traces shown in the descriptive drawing, Fig. 99, and are considered 
to indicate perfectly the path of the plane Q. 

101. Position of the Negative Trace. —The negative trace PS, 
in Fig. 96, is shown as one of the edges of the rectangular plate 
representing the unlimited plane P. This line PS has been de¬ 
termined by extending PH to meet the axis of Y (extended) at 










112 Engineering Descriptive Geometry 

b, and by extending PV to meet the axis of Z (extended) at c. 
The line joining b and c is the trace PS. It will be noted that in 
finding the location of PS in Fig. 97, PV has been extended to cut 
the axis of Z (extended up from ZO ) at c and PR has been ex¬ 
tended to cut the axis of Y (extended down from YO) at b. b 
has been rotated 90° about the origin, and the points b and c thus 
plotted (on § extended) have been found to give the line PS. 
Every step of the process and the lettering of the figure have been 
similar to those used in finding PS from PR and PV in Art. 98. 

In Eig. 98, the negative trace is QR, the top line of the rect¬ 
angular plate representing the unlimited plane Q. QR has been 
determined as follows: QV extended meets the axis of X extended 
at a, and QS extended meets the axis of V extended at b. The line 
ab is therefore the trace on ff, or QR. In the descriptive drawing 
the same process of extending QV to a and QS to b determines the 
line QR, a line every point of which has some negative coordinate. 
Of course QR must be considered as drawn on 2 iarts of the plane 
fi extended over V> S, and the so-called construction sjiace. In 
finding the negative traces, it is imperative to letter the diagrams 
uniformly, keeping, a for the intersection of the plane with the axis 
of X, b for that with the axis of Y, and c for that with the axis of 
Z. With this rule b will always be the point which is doubled by 
the separation of the axis of Y into two lines, and the arc bb will 
always be described in the construction space or in the quadrant 
devoted to V? never in those devoted to and §. 

102. Parallel Planes.—If two planes are parallel to each other, 
their traces on fj, V and S are parallel each to each. This prop¬ 
osition may be proved as follows: If we consider two planes P 
and Q parallel to each other and each intersecting the plane fj, the 
lines of intersection with ff (PR and QR) cannot meet, for, if 
they did meet, the planes themselves would meet and could not then 
be parallel planes. PR and QR must therefore be parallel lines 
described on ff. Thus, if a plane P and a plane Q are parallel, 
then PR and QR are parallel, PV and QV are parallel, and PS 
and QS are parallel. 

The method of finding the true length of a line by its projection 
upon a plane parallel to itself, treated in Chapter III, is really the 


Planes of Unlimited Extent: Their Traces 113 

process of passing a plane parallel to a projector-plane of the given 
line. Thus in Fig. 21, Art. 25, the auxiliary plane HJ has its hori¬ 
zontal trace XR parallel to A h B h , and the vertical trace of the ff 
projector-plane, if drawn, would be parallel to XK V . 

103. The Plane Containing a Given Line.—If a line lies on a 
plane, the trace of the line on any plane of reference (the point 
where it pierces the plane of reference) must lie on the trace of the 
plane on that plane of reference. Thus, if the line EF. Fig. 100, 
lies on the plane P, then A, the trace of EF on ff, lies on PR, the 
trace of P on J-j ; and B, the trace of EF on V? lies on PY, the 
trace of P on V- 



Y] 


Fig. 100 . 


Fig. 101 . 


From this fact it follows that to pass a plane which will contain 
a given line it is necessary to find two traces of the line and to pass 
a trace of the plane through each trace of the line. As an infinite 
number of planes may be passed through a given line, it is neces¬ 
sary to have some second condition to define a single plane. For 
example, the plane may be made also to pass through a given point 
or to be perpendicular to a reference plane. 

In Fig. 100, if only the line EF is given and it is required to pass 
a plane P, containing that line, and containing also some point, 
as a, on the axis of X, the process is as follows: Extend the line 
EF to A and B, its traces on H and V- Join B a an( ^ #-1. These 













114 


Engineering Descriptive Geometry 


i 


are the traces of the required plane P. In the descriptive drawing, 
Eig. 101, the corresponding operation is performed. A and B 
must be determined as in Art. 90, and joined to a. These lines 
represent the traces of a plane containing the line EF and the 
chosen point a. 

To pass a plane Q containing the line EF and also perpendicular 
to H (Figs. 100 and 101), the trace of Q on ff must coincide with 
the projection of EF on H, for the required plane perpendicular to 
ff is the fi projector-plane of the line. Its traces are therefore 
ABn and B h B. 

The traces of a plane containing EF and perpendicular to V are 
BA V and A V A. 

104. The Line or Point on a Given Plane. —To determine whether 
a line lies on a given plane is a problem the reverse of that just 
treated. It amounts simply to determining whether the traces of 
the line lie on the traces on the plane. Thus, in Fig. 101, if PV 
and PH are given, and the line EF is given by its projections, the 
traces of EF must be found, and if they lie on PH and PV the line 
is then known to lie on the given plane P. 

To determine whether a given point lies on a given plane is 
almost as simple. Join one projection of the point with any point 
on the corresponding trace of the plane. Find the other trace of 
the line so formed, and see whether it lies on the other trace of the 
given plane. Thus in Fig. 101, if the traces PH and PT T and the 
projections of any one point, as E , are given, select some point on 
PlI, as A, and join E fl A and E V A V . Find the trace B. If it lies 
on PV, the point E itself lies on P. 

To draw on a given plane a line subject to some other condition, 
such as parallel to some plane of reference, is always a. problem in 
constructing a line whose traces are on the traces of the given 
plane, and which yet obeys the second condition, whatever it may be. 

105. The Plane Containing Two Given Lines. —From the last 
article, if a plane contains two given linosj the traces of the plane 
must contain the traces of the lines themselves. The given lines 
must be intersecting or parallel lines, or the solution is impossible. 

In Fig. 102 two lines, AB and AC, are given by their projections. 
They intersect at A , since Ai,, the intersection of the fj projections, 


Planes of Unlimited Extent: Their Traces 


115 


is vertically above A v , the intersection of the V projections. Ex¬ 
tend the lines to E, F, G and H, their traces on ff and y. Join 
the ff traces, E and G, and produce the line also to a on the axis 
of X. Join the V traces, H and F, and extend the line HF alsG 
to a, Ea\ and aH are the traces of a plane P containing both lines, 
AB and AG. The meeting of the two traces at a- is a test of the 
accuracy of the drawing. 

This process may be applied to a pair of parallel lines, but not of 
course to two lines which pass at an angle without meeting. 



106. The Line of Intersection of Two Planes. —If two planes 
P and Q are given by their traces, their line of intersection must 
pass through the point where the Jffl traces meet and the point 
where the V traces meet. Thus, in Fig. 103, PH and QH meet 
at A and PV and QV meet at B. A and B are points on the 
required line of intersection of P and Q, and since A is on ff and 
B is on Y, they are the ff and y traces of the line of intersection. 
AB), and BA V are therefore the projections, and should be marked 
PQn and PQ V . 

















116 


Engineering Descriptive Geometry 


I 



Fig. 103. 



Fig. 104. 















Planes of Unlimited Extent: Their Traces 


117 


107. Special Case of the Intersection of Two Planes: Two Traces 

Parallel. —The construction must be varied a little in the special 
case when two of the traces of the planes are parallel. In Fig. l()d 
the traces PV and QV are parallel. In carrying out the construc¬ 
tion as in Fig. 100, it is necessary to join A v with B. But the 
point B is the intersection of PV and QV, which are parallel, and 
therefore is a point at an infinite distance in the direction of those 
lines, as indicated by the bracket on Fig. 104. To join A v with B 
at infinity means to draw a line through A v parallel to PV and QV. 


p„M Yk 


QH 

% 


C"\ \ 


A \ 

x o 

PQv 

> 

l 



PV 

A s 

QV Y z 



Ftg. 105. 


From B, at infinity, a perpendicular must be supposed to be drawn 
to the axis of X, intersecting it at Bh. Bn is therefore at an infinite 
distance to the right on the axis of X (extended). To join the 
point ,1 with the point Bh means, therefore, to draw a line through 
A parallel to the axis of X. These lines are the required projec¬ 
tions of PQ. 

108. Special Case of the Intersection of Two Planes: Four 
Traces Parallel. —Another special case arises when the four traces 
(on two planes of projection ) are parallel. It is then necessary to 
refer to a third plane of projection. In Fig. 105 the planes P and 
9 










118 


Engineering Descriptive Geometry 


I 


Q have their four traces on H and Y all parallel. The planes are 
inclined planes perpendicular to §, and if their traces are drawn 
on §, their intresection is the line PQ. In § both P and Q are 
“ seen on edge,” so their line of intersection is “ seen on end.” 
From PQ S , PQ V and PQn are drawn by projection. 

109. The Point of Intersection of a Line and a Plane. —The 
simple cases of this problem have been previously explained and 
used. If the plane is horizontal, vertical or inclined, there is 



always one view at least in which it is seen on edge. In that view 
the given line is seen to pierce the given plane at a definite point 
from which, by the rules of projection, the other views of the point 
of intersection are easily determined. Thus in Fig. 27, Art. 38, 
the point a, where PA pierces the plane KL, is determined first in 
V and then projected to H and §. 

The general case of this problem may be solved as in Fig. 106. 
A plane P is given by its traces PH and PV. A line AB is given 
by its projections. It is required to find where AB pierces P. The 








Planes of Unlimited Extent: Tiieih Traces 


119 


solution is as follows: Let a plane perpendicular to V be passed 
through the projection A V B V . According to Art. 103 the traces of 
this plane are B V F V and F V F. Draw the line of intersection of this 
plane with the plane P (Art. 106) as follows: B V F V and PV in¬ 
tersect at E . F and E are the traces of the line of intersection of 
the two planes. Complete the drawing of the line of intersection 
in ff, as FE h . 

Referring to the horizontal projection, AnB h is seen to intersect 
FEh, the H projection of the line of intersection, at Wh. Since 
both FE and AB are lines which lie in the vertical projector-plane 
through AB, this point of intersection, W h , is the projection of the 
true point of intersection, W, of those two lines. From Wu project 
to W v for the other projection of W. This point W which lies on 
P and is on the line AB is the required point. 

Problems XI. 

(For blackboard or cross-section paper or wire-mesh cage.) 

110. Plot the point A (4,7,9). Pass a horizontal plane P 
through the point A, and draw the traces of P. Pass a vertical 
plane Q, parallel to \, and draw its traces. Pass an inclined plane 
R, perpendicular to ff, making an angle of 45° with OX. 

111. Plot the line A (8,2,4), B (2,6,16). Pass an inclined 
plane P perpendicular to ff through this line and draw the traces 
of P. At C, the middle point of AB, pass a plane Q perpendicular 
to P and to Jj, and draw QII and QY. 

112. The plane P cuts the axes at the points a (10,0,0), 
1) (0, 5, 0) and c (0, 0,15). Pass a plane Q parallel to P, through 
the point a' (6, 0, 0). 

113. A plane P has its trace on Ifil through the points 
A (12,12,0) and & (0, 6, 0). Its trace on V passes through the 
point c (0, 0, 12). Draw the three traces. Draw three traces of a 
plane Q, parallel to P through the point c' (3, 0, 0). 

114. An indefinite line contains the points A (11,2,6) and 
B (5,6,0). Pass a plane P perpendicular to H containing this 
line and draw the traces PH, PV and PS. Pass a plane Q con¬ 
taining this line and the point a' (2,0,0). Draiv the traces QII 
and QV. Draw the negative trace QS on § extended over Jfl. 


120 


Engineering Descriptive Geometry 


I 


115. A plane P cuts the axis of X at a (4, 0, 0), the axis of Y 
at b (0, 6, 0), and the axis of Z at c (0, 0,-12). Draw its traces. 
Draw the V and § traces of a plane Q parallel to P and containing 
the line A (1, 4, 11), B (4, 1, 14). 

11G. An inclined plane, perpendicular to H, has for its V and 
§ traces lines parallel to OZ at positive distances of 15 and 5 units. 
An inclined plane Q perpendicular to ff has its V and § traces 
parallel to OZ at distances of 12 units and 8 units. Draw all three 
traces and the projection of PQ, their line of intersection. 

117. Draw the traces of a plane P, containing the points 

A (8,1,.3), B (4, 5,1) and C (2, 4, 3). Does the point D (4, 1, 5) 
lie on this plane? 

118. The traces of a plane P are lines through the points 

a (10,0,0), 1) (0,15,0) and E (14,0,6). A plane Q has its 

traces through the points a! (2,0.0), E, and F (7,5,0). Draw 

the projections of their line of intersection, PQ. 

119. The plane P cuts the axes at a (12,0,0), 1) (0, 12,0) and 
c (0,0,12). Where does the line E (1,5,12), F (5,3,6) pierce 
the plane? 


CHAPTER XII. 

VARIOUS APPLICATIONS. 


110. Traces of an Inclined Plane Perpendicular to an Oblique 
Plane. —One of the most general devices used in the drafting room 
is the auxiliary plane of projection, and it is often advantageous 
to pass this plane perpendicular to some plane of the drawing in 



order to get the advantage of showing that plane “ on edge.” Thus 
in Fig. 31, Art. 42, the plane U has been taken perpendicular to 
the long rectangular faces of the triangular prism, in order to 
show clearly where BB' and DD' pierce those planes. The manner 
of passing the plane JJ was fairly clear in that case from the 
simplicity of the figure. However, as it is not always clear how to 
pass a plane perpendicular to an oblique plane, the general method 
may well he explained here. In Fig. 107 the plane P, previously 
shown in Fig. 94, is represented, and an auxiliary plane 5J> per¬ 
pendicular to it and to H, is shown. The traces of P are PH, PV 
and PS as before, and the traces of HJ are U1I and US. It must 













122 


Engineering Descriptive Geometry 


( 


be understood that the ff traces of these planes, PH and UH, are 
perpendicular to each other, as this condition is essential if P and 
U are to be planes perpendicular to each other. 

Eig. 108 is the descriptive drawing corresponding to the per¬ 
spective drawing, Fig. 107. At some point h on PH a line Mdh 
has been drawn perpendicular to PH. This line is the inclined 
trace of a plane UJ perpendicular to UJ. The other traces of UJ are 
parallel to the axis of Z (Art. 98). One of these, the trace on §, 
is shown by the line d s N s , parallel to OZ, d h and d s being two 
representations of the same point d in Fig. 107, just as bn and b s 
represent the point b, duplicated. Mdn may be called UH and d s N s 
may be called US. UH and US are the traces of an inclined plane 
U, perpendicular to the oblique plane P. 

The proof that P and UJ are perpendicular to each other is as 
follows: If, in Fig. 107, a line hh' is drawn perpendicular to UJ 
at the point h, it will lie in the plane UJ. The angle ahld will 
then be an angle of 90°, and by construction the angle ahd is also 
90°. Thus the line ah is perpendicular to two intersecting lines de¬ 
scribed in the plane UJ and is therefore perpendicular to UJ itself. 
The plane P contains the line PH and is thus perpendicular to UJ* 

111. An Auxiliary Plane of Projection Perpendicular to an 
Oblique Plane. —To utilize the inclined plane UJ as an auxiliary 
plane of projection, its developed position must be shown by drawing 
di,N u perpendicular to UIP. This line is the duplicate position of 
d 8 N 8 or US. In developing the planes, UJ is first revolved on UH 
as an axis into the plane of H as shown in Fig. 109, and then with 
UJ into the plane of the paper, V- The trace of P on UJ, or PU, is 
the line of intersection of the planes, and is shown clearly in Fig. 
107. This line passes through h where PH and UII meet, and 
through s where PS and US meet. In Fig. 108, dj,s is laid off on 
dhN u , equal to d s s, and the line hs is the required trace of P on JJ? 
or PU. The actual line PU, in Fig. 108, is only that part of PU, 
in Fig. 107, which is between h and s, shown as a broken line. 

The important part in this process is that UJ is taken perpen¬ 
dicular to P, so that P is “ seen on edge ” on UJ. By this process 
the plane P, which is oblique when JJ, V arid § are considered, 


Various Applications 


123 


becomes an inclined plane when only ff ancl JJ are considered. 
As it is easier to deal with inclined than with oblique planes, we 



z 

Fig. 110. 


may now treat P as inclined to H and perpendicular to U in 
further operations. 

Fig. 108 is well adapted to making a paper box diagram which, 






































124 


Engineering Descriptive Geometry 


I 


when folded, will give most of the lines of Fig. 107. To reconstruct 
Fig. 108, plot the points a (18,0,0), b (0,18,0), c (0,12,0), 
d (0,6,0), h (6,12,0) and s (0,6,8). The line d h A J tl is at an 
angle of 45° with ZOY h and the construction space Y s Odi,N u can 
be folded away inside by creasing or cutting it on several lines. 

112. Intersection of an Oblique Plane and a Cylinder. —An ex¬ 
ample of the use of an auxiliary view on which an oblique plane is 
seen on edge is shown in Fig. 110. An inclined c} r linder is inter¬ 
sected by an oblique plane P given by its traces PH, PV and PS. 
It is required to describe on the cylinder the curve of intersection 
of the plane and the cylinder. The solution is as follows: An 
auxiliary plane 5J, perpendicular to P and to H? is chosen, and 
PU is drawn upon HJ as in Fig. 108. PU is the view of P “ seen on 
edge ” in U- Auxiliary cutting planes parallel to H are used for 
the determination of the required line of intersection. The traces 
of one of the planes are drawn, as T'T in V? FT" in §, and T"T"" 
in U- This latter trace is parallel to d h M (or UH), because T is 
parallel to H, and the distance d h T" is equal to d s T" in g. T'T'" 
cuts the axis of the cylinder at p. p is projected to J-J, and the 
circular element described in fj, with p as a center, is the inter¬ 
section of the auxiliary plane T and the cylinder. In UJ the planes 
P and T are both seen “ on edge,” intersecting in a line seen on end. 
This point projected to H gives this line of intersection of P and T 
as tt'. 

The intersections of the intersections are therefore the points t 
and f, where the circle and the straight line meet. 

113. The Angle between Two Oblique Lines.—This problem of 
finding the angle between two oblique lines is shown in Fig. 111. 
Let two lines AB and AC, meeting at A, be given by their and 
V projections. It is required to find the true angle between them. 

By the process of Art. 105, Fig. 102, the traces of the plane con¬ 
taining AB and AC are found and the lines are all lettered accord¬ 
ing to Fig. 102. 

An auxiliary plane of projection, is passed perpendicular to 
PV, and therefore perpendicular to both P and V, and is revolved 
into the plane V- The projections of AB and AC on this plane 



Various Applications 


125 


fall in the single line A U C U B U , since P, the plane of the lines, is 
“ seen on edge ” on JJ. A portion of the plane P is now revolved 
about the UJ projector of the point A into a position parallel to 
XM. In U, C u moves to C' u and B u to B' u , revolving about A as 
their center. In V, B v moves to B' v and C v to C' v , both parallel to 
XM. This is the process of finding the true length of a line by 



revolving about a projector, as in Art. 32. A V B' V is the true length 
of AB; A V C' V is the true length of AC; and B' V A V C' V is the true 
angle between the lines. 

This process makes it possible to find the true shape of any 
figure described on an oblique plane. 

114. A Plane Perpendicular to an Inclined Line. —It is often 
advantageous to pass a plane perpendicular to a line in order to 
use the plane as a plane of projection, on which the given line will 
be seen on end as a point. The method of passing a plane perpen- 









12(3 Engineering Descriptive Geometry 

dicular to an inclined line is shown in Eig. 112. Let AB be an 
inclined line, lying in a plane parallel to \, so that Ai,Bh is parallel 
to the axis of X. It is required to find the traces of a plane P 
perpendicular to AB. The essential point is that the traces of the 
plane must be perpendicular to the corresponding projections of 



the line. Thus, choose some point p on the inclined projection of 
the line, in this case on A V B V , and through p draw a perpendicular 
to A V B V , to serve as the trace of P. At a, where this trace PV 
meets the axis of X, erect a perpendicular to PH. These lines PV 
and PH are the traces of an inclined plane perpendicular to AB 
and to V- It is noticeable that the inclined trace of the plane is 






Various Applications 


127 


on that reference plane which shows the inclined projection of the 
line.* 

115. Application of a Plane Perpendicular to a Line. —In Fig. 
113 an application of an inclined plane perpendicular to an in¬ 
clined line is made for the purpose of finding the line of intersection 
between an inclined cone and an inclined cylinder whose axes do 
not meet. 



If from P, the vertex of the cone, a line Pp is drawn parallel to 
QQ ', as shown, any plane which contains this line and cuts both 

*A proof that P is perpendicular to AB is as follows: AB 
is the line of intersection of its own fi projector-plane, and its 
own V projector-plane. P is perpendicular to both these projector- 
planes. For, P is perpendicular to V and therefore to the ff pro¬ 
jector-plane, w r hich is parallel to V i the V projector-plane is per¬ 
pendicular to V, so that it is seen on edge on V just as is P itself; 
apA v is therefore the true angle between these two planes, and by 
construction is a right angle. P is therefore perpendicular to both 
projector-planes and therefore to the line AB. which is their line 
of intersection. 















































128 


Engineering Descriptive Geometry 


i 




surfaces will cut only simple elements of the surfaces. For such 
a plane contains the vertex of the cone, and therefore, if it cuts 
the cone, will cut it in straight elements; and such a plane is 
parallel to QQ' , and therefore, if it cuts the cylinder, cuts only 
straight elements. Xo other planes can be found which cut simple 
elements and can be used to determine the line of intersection. 

If a plane U is passed perpendicular to Pp at any point p, and 
is used as an auxiliary plane of projection, Pp will be seen on end 
as the point P, and any plane Pi through P seen on edge in JJ, as 
PR ', will cut only straight elements on the two curved surfaces. 
The complete projections of the cone and cylinder have been shown 
on UJ> and the plane R cuts the bases at a, ~b, c and d. These points 
projected to V enable the elements to be drawn there, and the 
intersections of the intersections are the four points marked r. 
From V these points are projected to H and §. Two of these 
points r have been projected to the other views to show the neces¬ 
sary construction lines. 

116. A Plane Perpendicular to an Oblique Line.— To pass a 

plane perpendicular to an oblique line, it is only necessary to draw 
the traces of the plane perpendicular to the corresponding pro¬ 
jections of the line. In Fig. 114, let AB be an oblique line. At 
any point on A }l B } , draw a perpendicular line PH. From a;, where 
PH meets the axis of X, draw PV perpendicular to AB* 

A paper box diagram traced from Fig. 114, or constructed on 
coordinate paper, using the coordinates A (10, 4, 4) and B (6, 8, 2), 
C (2,12, 0) and D (14, 0, 6), and a (8, 0, 0), will assist materially 
in understanding the problem. 

The oblique plane P is not serviceable as an auxiliary plane of 
projection. 

117. The Application of Axes of Projection to Mechanical 
Drawings. —Descriptive Geometry is a geometrical science, the 
science dealing primarily with orthographic projection, while Me¬ 
chanical Drawing is the art of applying these principles to the 

* The proof of this construction is more difficult than in the 
corresponding case of an inclined line, but it depends as before 
on the line AB being the intersection of its H and V projector- 
planes, and these planes themselves being perpendicular to P. 




4 


Various Applications 


129 


needs of engineers and mechanics in the pursuit of industries. 
Mechanical Drawing includes therefore many abbreviations and 
conventional representations, which seek to curtail unnecessary 
work and often to convey information as to methods of manu¬ 
facture and other such commercial considerations foreign to the 
strict scientific study. 



In Mechanical Drawing many lines necessary to the strict execu¬ 
tion of a descriptive drawing are omitted as unnecessary to the 
application of the principles, when once the principles have been 
fully grasped. A noteworthy omission is the axes of projection, 
which, though absent, still govern the rules for making the draw¬ 
ing. Instead of measuring distances from the axes for every point 











130 


Engineering Descriptive Geometry 


l 


on the drawing, the “ center lines ” of the different views (which 
realty represent central planes) are laid off and distances from 
these center lines are thereafter used. This is the regular pro¬ 
cedure in drawing-room practice. That this difference is purely 
one of omission is clear from the fact that axes of projection may 
always be inserted in a mechanical drawing. If two views only of 
a piece are presented, any line between them (perpendicular to the 
lines of projection from one view to another) may be selected as 
the axis of X, and any convenient point on that line as the origin 
of coordinates. 

If three views are given, as, for example, Fig. 32, Art. 44, sup¬ 
posing the axes to be there omitted, a ground line XOY s may be 
selected at will, dividing the fields of JH} and V- The other line 
must be determined as follows: By the dividers take the vertical 
distance from OX to the center line mn, and lay off this distance 
horizontally to the left from the center line of the side elevation. 
The line ZOYn may be drawn. All y coordinates of points will 
now check correctly, measured parallel to the two axes'of Y, if the 
original drawing itself is accurate. 

It is thus evident that in applying Descriptive Geometry to prac¬ 
tical mechanical drawing we may fall back upon the use of axes of 
projection whenever the lack of them is felt. 

118. Practical Application of Descriptive Geometry. —Many 
draftsmen have picked up a knowledge of Descriptive Geometry 
without direct study of the science. This is largely due to the fact 
that, till very recentty, all books on Descriptive Geometry were 
based on a system of planes of projection which are analogous to 
the methods of practical drawing in use on the continent of Europe, 
but which are liftle used in England, and hardly at all in the United 
States of America. It will be found, however, that in American 
drafting rooms all the usual devices of draftsmen are applications, 
sometimes almost unconscious applications, of the principles covered 
in the preceding chapters. The favorite device is the application 
of an inclined auxiliary plane of projection, suitably chosen; next 
in importance is the rotation of the object to show" some true shape ; 
while other applications are used less frequently. The methods of 
determining lines of intersection of planes and curved surfaces are 
exactly those described in Chapters IV, VII and VIII. 


Various Applications 


131 


Problems XII. 

(For use on blackboard, with cross-section paper or wire-mesh 

cage.) 

120. The plane P has its traces through the points a (14, 0, 0), 
b (0^14,0) and c (0,0,7). Pass a plane Q, perpendicular to P 
and to fl, through the point A (5, 7, 0). If Q is to be used as an 
auxiliary plane of projection, draw the trace of P on Q when Q has 
been revolved into coincidence with JJ. 

121. Draw the traces of a plane P cutting the axes at the points 
a (12,0,0), b (0,8,0) and c (0,0,12). Draw the traces of an 
auxiliary plane, JJ, perpendicular to PH at the point A (3, 6, 0). 
Is the point B (6,1, 4J) on the plane P ? 

122. The J-J trace of a plane P passes through the points 
A (12, 5, 0) and B (6,2, 0). Its V trace passes through C (9, 0, 6). 
Pass an inclined plane perpendicular to Jf-f and perpendicular to 
P, through the point D (5, 9, 7). 

123. Of a plane P, TIP, the horizontal trace, passes through the 
points A (5,3,0) and B (13,9,0), and VP passes through 
C (12, 0, 11). Complete the traces of P and draw the traces of a 
plane perpendicular to VP at the point D (8, 0, 8). Prove that the 
line B (9, 6,1), F (6, 3, 2) lies on the plane P. 

124. A sphere of radius 7 units has its center at C (8, 8, 8). A 
plane P cuts the axes of projection at a (26, 0, 0), b (0,13, 0) and 
c (0,0,13). Pass an auxiliary plane of projection JJ, perpen¬ 
dicular to H and to P, cutting the axis of X at d (16, 0, 0). Draw 
the trace of P on HJ- The circle of intersection of the sphere and 
the plane P is seen on edge on \J. Show the elliptical projection 
of this circle, on ff, by passing auxiliary cutting planes parallel 
to U- (If this problem is solved by use of wire-mesh cage, the 
point a is inaccessible, but PTI passes through E (16,5,0), and 
PV through F (16,0,5). The plane S' can be turned to serve 
as U-) 

125. Find the true shape of the triangle A (3, 2, 6), B (9, 6, 2), 
C (8, 0, 0). Find the traces of two of the sides of the triangle and 
pass the plane UJ perpendicular to the plane of the triangle and 
perpendicular to ff, and through the point D (0,7,0). 


132 


Engineering Descriptive Geometry 


I 


126. Find the true shape of the triangle A (7, 6, 1), B (4, 2, 9), 
C (10,2,3). Find the traces of two of the sides of the triangle 
and pass the plane UJ perpendicular to the plane of the triangle 
and perpendicular to JJ, and through the point D (0, 1, 0). 

127. Draw the traces of a plane P perpendicular to V and to 
the line A (2,6,9), B (8,6,5) at C (11,6,3). If this plane is 
used as an auxiliary plane of projection, what is the projection of 
AB on it? 

128. Draw the traces of a plane P perpendicular to H and to 
the line .4 (3, 9, 6), B (13, 4, 6), at C (17, 2, 6), a point on AB. 
(If wire-mesh cage is used for the solution, turn §' to serve as UJ 
and draw on it the view of A U B U .) 

129. Draw the three traces of a plane P perpendicular to the 
oblique line A (8, 12, 5), B (14, 3, 7). Show that all three traces 
are perpendicular to the corresponding projections of AB. 


CHAPTER XIII. 


THE ELEMENTS OF ISOMETRIC SKETCHING, 

119. Isometric Projection. —There is one special branch of 
Orthographic Projection which is of peculiar value for represent¬ 
ing forms which consist wholly or mainly of plane faces at right 
angles to each other. Ordinary orthographic views are projec¬ 
tions upon planes parallel to the principal plane faces of the object, 
as shown in Eig. 2, Art. 4. If, however, instead of the regular 
planes of projection, the object is projected upon a new plane of 
projection, making the same angle with each of the regular planes, 
an entirely different result is obtained, called an isometric pro¬ 
jection A This view has the useful property that it has all the air 
of a perspective and may, with certain restrictions, be used alone 
without other views us a full representation of the object. 

In Art. 21 the method of converting the perspective drawings 
of this treatise into isometric sketches was explained in a rough 
and unscientific way. In this chapter there is explained the method 
of making isometric sketches from models, as a step to making 
orthographic drawings or isometric drawings. 

120. Isometric Sketches of Rectangular Objects. —Figs. 19 and 
19a are the isometric drawings of a cube. Since the line of sight 
from the eye to the point 0 makes equal angles with ff, V and §, 
the three planes must subtend the same angle at 0. XOY . YOZ 
and XOZ are each 120°. though representing angles of 90° on the 
cube. Since opposite edges of ff are parallel, it follows that each 
face of the cube is a rhombus and that the cube appears as a regular 
hexagon, all edges appearing of exactly the same length. This 
fact is the basis of the name “ isometric,” meaning “ equal- 
measured.” 

Figs. 115, 11G and 117 are sketches of other objects, all of whose 
corners are right angles. The angles at these corners appear there¬ 
to 


Engineering Descriptive Geometry 


l 


134 


fore like those of the cube, either as 60° or 120° on the isometric 
sketch. 

In making the isometric sketch from a model having rectangular 
faces, the first step is to put the object approximately in the iso- 



Fig. 115. 



Fig. 116. 


Fig. 117. 



Position for 
Orthographic 
Projection . 

Fig. 118. 



Turned 4-5° 
about a verti¬ 
cal axis. 


Fig. 119. 



about an hori¬ 
zontal axis. 


Fig. 120. 


metric position. At any projecting corner imagine a line to project 
from the corner so as to make equal angles with the three edges 
which meet at the given corner. View the object by sighting along 
this imaginary line and begin the sketch from that view. 



































The Elements of Isometric Sketching 


135 


If there is any difficulty in finding this line of vision directly, 
the object may be turned horizontally through, an angle of 45° and 
tilted down through an angle of 35° 44'. This operation is the 
basis of the method of finding the “ isometric projection.” 

Figs. 118, 119 and 120 show the steps in passing from the ortho¬ 
graphic position to the isometric position, the model used being a 
rectangular block with a lengthwise groove cut in one face. 

121. Isometric Axes.— It will be noticed in the previous iso¬ 
metric figures that all lines are drawn in one of three general direc¬ 
tions. One of these directions is usually taken as vertical and the 
other two directions make angles of 120° with the vertical. These 
three directions are known as the isometric axes. In this sense 
the word axis means a direction, not a line. 

In plotting points from a selected origin, the x coordinates are 
plotted up and to the left, the y coordinates up and to the right, 
and the 2 coordinates vertically downward, as in Fig. 19a. 

122. Isometric Paper. —Paper ruled in the direction of the iso¬ 
metric axes is called isometric paper, and is of great assistance in 
making isometric sketches. The lines divide the paper into small 
equilateral triangles. 

In sketching, the sides of these equilateral triangles are taken to 
represent unit distances, exactly or at least approximately. Thus, 
if the model shown in Fig. 120 is a block 3" x 3" x 3 V , with a 2" x 1" 
groove lengthwise along one face, some point a on the paper is 
selected, and from it distances are taken along the isometric axes, 
so that each unit space represents one inch. 

From a three units are counted vertically downward, eight up, 
and to the right, and one unit, followed by a gap in the line of one 
unit, and then a second unit, up to the left. Thus all lines of the 
sketch follow the ruled lines as long as the dimensions of the model 
are in even inches. 

An isometric sketch made in this manner, particularly if spaces 
have been exactly counted off according to the dimensions of the 
piece, is practically an isometric drawing. If fully dimensioned, a 
sketch on plain paper proportioned by the eye is nearly as good as 
one in which spaces are counted exactly. Such sketches serve all 


136 


Engineering Descriptive Geometry 


I 


purposes, though of course more difficult to make than those on 
isometric paper. 



123. Non-Isometric Lines in Isometric Sketching. — Objects 

which have a few faces and edges oblique to the principal plane 
faces may still be shown by isometric sketching. In such cases it is 
always well to circumscribe a set of rectangular planes about the 



oblique parts of the object to aid the imagination. Dimension 
extension lines should be used for this purpose. In using isometric 
paper this squaring up is done by the lines of the paper. 




























































The Elements of Isometric Sketching 


137 


Figs. 122, 123 and 124 are good examples of oblique lines and 
faces. Figs. 123 and 124 show also the circumscribed isometric 
lines which “ square up ” the oblique parts. 

124. Angles in Isometric Sketching.—In isometric sketching 
angles do not, as a rule, appear of their true magnitude. Thus the 
90° angles on the faces of the brick appear in Fig. 115 as 60° or 
120°, but not as 90°. In general, the lengths of oblique or inclined 
lines depend on position, and are not subject to measurement by 
scale. 

The lines which square up oblique parts are useful in giving the 
tangent of the angle of an oblique surface. Thus in Fig. 124, the 
angle a differs in reality from the angle as it appears in either place 

marked, but the tangent of a is u . In Fig. 123, 0 = tan _1 — . In 

v n 

practice angles are often given by their tangents. Thus the slope of 



a roof is given as “ one in two ” or the gradient of a railroad as 
“ three per cent/’ 

125. Cylindrical Surfaces in Isometric Sketching.—In ortho¬ 
graphic drawings circles appear commonly on planes parallel to the 
three planes of projection. To illustrate the position and appear¬ 
ance of circles in isometric drawing in the three typical cases, Fig. 
125 represents the isometric sketch of a cube, having a circle in¬ 
scribed in each square face. 

Each of the faces of the cube is perpendicular to the isometric 
axis given by the intersection of the other two faces. Thus the 
square ABCD is perpendicular to the edge BF. The circle abed, 














138 


Engineering Descriptive Geometry 


I 


inscribed in the square ABCD, appears as an ellipse, whose minor 
axis, ef, lies on the diagonal BD of the square, BD appearing as a 
continuation of the edge FB. In all three cases, then, the minor 
axis of the ellipse lies in the same direction, on the sketch, as that 
isometric axis to which the plane of the circle is in reality perpen¬ 
dicular. 

The major axis is necessarily perpendicular to the minor axis, 
and lies on the other diagonal of the square. 

Since the cylinder is the surface most used in engineering, the 
rule may be applied to cylinders as follows: The ellipse which 
represents the circular base of any cylinder must be so sketched 
that its minor axis is in line with the axis or center line of the 
cylinder. Fig. 126 is an isometric sketch of a piece composed of 
cylinders. All the ellipses are seen to follow this rule. 

In sketching cylindrical parts of objects, it is necessary to im¬ 
agine them squared up by the use of isometric lines and planes. 
Thus the first steps in sketching the piece of Fig. 126 are shown 
in Fig. 127. The circumscribing of a square about a circle in the 
object corresponds to circumscribing a rhombus about the ellipse 
in the isometric sketch. It now remains to inscribe an ellipse in 
the rhombus. This ellipse must be tangent to the rhombus at the 
middle of each side. To sketch the ellipse, as for example the small 
end in Fig. 127, draw the diagonals of the rhombus to get the 
directions of the major and minor axes, and find the middle points 
of the sides (by center lines, through the intersection of the diagon¬ 
als) . It is now easy to sketch the ellipse, having four points given, 
the direction of passing through those points, and the directions of 
the major and minor axes. 

126. Isometric Sketches from Orthographic Sketches. —A good 

exercise consists in making isometric sketches from orthographic 
sketches or drawings. The three coordinate directions, x, y and 2 , 
must be kept in mind at all times. Fig. 128, as an example, is most 
instructive. From the orthographic sketches, Fig. 128, the iso¬ 
metric sketch, Fig. 129, is to be made. A point a is selected to rep¬ 
resent a point a on the orthographic views. The line ab is an x 
dimension and is plotted up to the left; ac is a y dimension, and is 
plotted up to the right; while ad is a 2 dimension, and is plotted 


The Elements of Isometric Sketching 


139 


vertically downward. The semicircle is inscribed in a half-rhombus, 
tangent at b, e and f. 






































































































c 









f 





b 












J 







( 

a, 















1 




> 

< 

9 







T 





% 


"i 

r 




























x 





J 





\ 

















s' 


< 
























_ 

— 

— 

— 

— 



»- 





















l > 

















i 

/ 

















\ 

~S— 






Fig. 128 . 


Fig. 129 . 


The cross-section lines of Fig. 128 and the isometric lines of Fig. 
129 are represented as overlapping between the figures. Some iso¬ 
metric paper is ruled in this manner, so that it may be used for 
both purposes. 























































































140 


Engineering Descriptive Geometry 


Problems XIII. 

(For blackboard or isometric paper.) 

130. Make an isometric sketch of the angle piece, Fig. 130, using 
the spaces for 1" distances. 




131. Measure the tool-chest, Fig. 131, and make a. bill of ma¬ 
terial, tabulating the boards used, and recording their sizes, giving 
dimensions in the order: width, thickness, length, thus: 

Mark. Name. Size. Number. 

A. Top of Chest. 14" x 1" x 34". 2. 

132. A parallelopiped, 9" X 6" X 3", has a 3" square hole from 
center to center of the largest faces, and a 2" bore-hole centrally 
from end to end. Make an isometric sketch. 

133. Let Fig. 3, Art. 5, represent a model cut from a 12" cube 
by removing the center, leaving the thickness of the walls 3". Let 
the angular point form a triangle whose base is 12" and altitude 8". 
Make an isometric sketch. 


















































The Elements oe Isometric Sketching 


141 


134. A cube of ICE has a 6" square hole piercing it centrally from 
one side to the other, and a 4" bore-hole piercing it centrally from 
side to side at right angles to the larger hole. Make an isometric 
sketch. 

135. A grating is made by nailing slats J"x^"xl2", spaced •£" 
apart, on three square pieces, 1J" square, 22" long, spaced 4-J" apart. 
Make an isometric sketch. 

136. Make orthographic sketches of the bracket, Fig. 122. Views 
required are plan and front elevation. (On cross-section paper use 
the unit distance for the unit of the isometric paper. On black¬ 
board let each unit of the isometric paper be represented by a dis¬ 
tance of 2".) 

137. Make isometric sketches of Fig. 11, Art. 14, and Fig. 24, 
Art. 32. 

138. Make isometric sketches of Fig. 13, Art. 15, and of Fig. 82, 
Art. 89. In Fig. 82 let A be the point (9, 8, 0) and B the point 
(9,0, 12). 

139. Make an isometric sketch of Fig. 71, Art. 84, the diameter 
of the cylinder being 7 units and the length 14 units. 

140. Make an isometric sketch of Fig. 92, Art. 93, using the 
coordinates given in Art. 94. 


CHAPTER X1Y. 

ISOMETRIC DRAWING AS AN EXACT SYSTEM. 


127. The Isometric Projection on an Oblique Auxiliary Plane.— 

The sketches previously considered have generally had no exact scale. 
Those drawn on isometric paper have a certain scale according to 
the distance which one unit space of the paper actually represents. 



If the isometric projection is derived from an orthographic draw¬ 
ing of the usual kind by the law T s of projection, the isometric projec¬ 
tion so formed has of course the same scale as the original drawing. 

In Fig. 132 an isometric projection of a cube is derived from the 
orthographic drawing by the use of an inclined plane of projection, 

























































































Isometric Drawing as an Exact System 


143 


U> and an oblique auxiliary plane of projection W- The aim is 
to produce the projection on a plane making the same angle with all 
three edges of the cube meeting at any one corner. This plane must 
be perpendicular to a diagonal of the cube. In Fig. 132 this di¬ 
agonal is the line EC, a true diagonal, passing through the center of 
the cube, not a diagonal of one face of the cube. 

The first, or inclined, auxiliary plane JJ is taken parallel to the 
V projection of EC, and therefore perpendicular to V and making 
an angle of 45° with ff and §. The projection of EC on UJ shows 
its true length. 

The second, or oblique, auxiliary plane is taken perpendicular 
to EC. It is oblique as regards H and V? but, as EC is a line par¬ 
allel to U> anc l 'W is perpendicular to EC, W is perpendicular to 
5J- As regards V and U, 'W" is an inclined plane, having its in¬ 
clined trace MN on ILL the trace on V being a line ML V , perpendic¬ 
ular to ZM, the trace of UJ on V- The construction of this second 
projection is therefore according to the usual methods. Any point, 
as F, is projected by a perpendicular line across the trace MN and 
the distance nF w is laid off equal to mF v . 

The projection on W is the isometric projection of the cube and 
is full-size if the plan and front elevation are full-size projections. 
The edges are all foreshortened, however, and measure only yyy °f 
their true length. 

128. The Angles of the Auxiliary Planes.—The plane UJ makes 
an angle of 45° with the plane fj. The plane VV" makes an angle 
of 35° 44' with §, or (90° — 35° 44') with V- If the side of the 
cube is taken as 1, the length of the diagonal of the face of the cube 
is V2, and the length of the diagonal of the cube is V3. The 


first angle is that angle whose tangent is -y , or whose sine is -i-. 

\/2 


1 


The second angle is that angle whose tangent is —= and whose sine 

\/2 


IS 


\/2 

x/3 


129. The Isometric Projection by Rotating the Object.—In Fig. 
134 is shown a method of deriving the isometric projection by turn¬ 
ing the object. The plan, front, and side elevations are drawn with 



144 


Engineering Descriptive Geometry 


v 


the object turned through an angle of 45° from the natural posi¬ 
tion (that in which the faces of the cube are all parallel to the 
reference planes). The side elevation shows the true length of one 
diagonal of the cube, AG. Some point on AG extended, as K, is 
taken as a pivot, and the whole object is tilted down through an 
angle of 35° 44', bringing AG into a horizontal position, A'G'. The 



Fig. 134. 


new projection of the object in V is the isometric projection. This 
process of turning the object corresponds to the turning of the 
object in isometric sketching, as shown in Figs. 118, 119 and 120. 

The isometric projection of the cube has all eight edges of the 
same length, but foreshortened from the true length in the ratio of 
V3 to \/2. 

Any object of a rectangular nature may be treated by either 
process to obtain the isometric projection. 

















































Isometric Drawing as an Exact System 


145 


130. The Isometric Drawing’. —To make a practical system of 
drawing capable of representing rectangular objects in an unmis¬ 
takable manner in one view, the fact that all edges are foreshort¬ 
ened alike is seized upon, but the disagreeable ratio of foreshorten¬ 
ing is obviated by ignoring foreshortening altogether. 

An isometric draiving is one constructed as follows: On three 
lines of direction, called isometric axes , making angles of 120° with 
each other, the true lengths of the edges of the object are laid off. 
These lengths, however, are only those which are mutually at right 
angles on the object. All other lines are altered in shape or length. 
An isometric drawing is distinct from an isometric projection, as 
it is larger in the proportion of 100 to 83 (V3: V2). The iso¬ 
metric drawing of a 1" cube is a hexagon measuring 1" on each 
edge. 

131. Requirement of Perpendicular Faces. —An isometric draw¬ 
ing, being a single view, cannot really give “ depth,' 1 '’ or tell exactly 
the relative distances of different points of the object from the eye. 
It absolutely requires that the object drawn shall have its most 
prominent faces, at least, mutually perpendicular. The mind must 
be able to assume that the object represented is of this kind, or the 
drawing will not be “ read ” correctly. Even on this assumption, 
in some cases isometric drawing of rectangular objects may be 
misunderstood if some projecting angle is taken as a reentrant one. 
Thus in Eig. 133 we have a drawing which might be taken as the 
pattern of inlaid paving or other flat object. If it is taken as an 
isometric drawing and the various faces are assumed to be perpen¬ 
dicular to each other, it becomes the drawing of a set of cubes. 
Curiously enough, it can be taken to represent either 6 or 7 cubes, 
according as the point A is taken as a raised point or as a depressed 
one. In other words, it even requires one to know just how the 
faces are perpendicular to each other to be able to take the drawing 
in the way intended. 

This requirement of perpendicular faces limits the system of 
drawing to one class of objects, but for that class it is a very easy, 
direct, and readily understood method. Untrained mechanics can 
follow isometric drawings more easily than orthographic drawings. 


Both fi I let5 Ij R 


146 


Engineering Descriptive Geometry 






















Isometric Drawing as an Enact System 


147 


132. The Representation of the Circle. —In executing isometric 
drawings, the circle, projected as an ellipse, is the one drawback to 
the system. To minimize the labor, an approximate ellipse must 
be substituted for an exact one, even at the expense of displeasing 
a critical eye. The system, if used, is used for practical purposes 
where beauty must be sacrificed to speed. In Fig. 125 the rhombus 
ABCD is the typical rhombus in which the ellipse must be inscribed. 
The exact method is shown in Fig. 43, but requires too much time 
for constant use. The following draftsman’s ellipse, devised to be 
exactly tangent to the rhombus at the middle point of each side, is 
reasonably accurate. From B, one extremity of the short diagonal 
of the rhombus, drop perpendiculars Bd and Be upon opposite 
sides, cutting the long diagonal at h and l. With B as a center and 
Bd as a radius, describe the arc dc. Similarly, with D as a center, 
describe the arc ba. With h and l as centers, and led as a radius, 
describe the arcs ad and cb. The resulting oval has the correct 
major axis within one-eighth of 1 per cent, and has the correct 
minor axis within 34 per cent. 

This draftsman’s ellipse is exact where required, namely, on the 
two diameters ac and db, wdiicli are isometric axes, and it is prac¬ 
tically exact at the extremity of the major axis. 

133, Set of Isometric Sketches. —Fig. 135 is a set of isometric 
sketches of the details of the strap end of a small connecting-rod, 
from which to make orthographic drawings. The isometric sketch 
is much clearer than the corresponding orthographic sketch, and 
the set shows clearly how the pieces are assembled. 

The orthographic drawing of the assembled rod end is much 
easier to make than the assembled isometric drawing. It is in fact 
clearer for the mechanic than the assembled isometnc drawing 
would be, for the number of lines would in that case be quite con¬ 
fusing. It illustrates well the fact that isometric sketches and 
drawings should be limited to fairly simple objects. 

Another noteworthy fact is that center lines, which should always 
mark symmetrical parts in orthographic drawings, should be used 
in isometric drawing only when measurements are recorded from 
them. 

The sketch as given is taken directly from an examination paper 
used at the IT S. Naval Academy for a two-hour examination. On 


148 


Engineering Descriptive Geometry 


account of the shortness of the period, only one orthographic view, 
the front elevation, is required, but if time were not limited, a plan 
also should be drawn. 

The following explanation of the sheet is printed on the original: 

“ Explanation of Mechanism .—The isometric sketches represent 
the parts of the strap end of a connecting-rod for a small engine. 
In assembling, A, B, C, and D are pushed together, with the thin 
metal liners, G, filling the space between B and C. The tapered 
key, E, is driven in the J" holes of A and D, which will be found to 
be in line, except for a displacement of which prevents the key 
from being driven down flush with the top of the strap D. The two 
bolts, F. are inserted in their holes, nuts 77 screwed on, and split 
pins (which are not drawn) inserted in the -J" holes, locking the 
nuts in place. In time the bore of the brasses B and C wears to 
oval form. To restore to circular form, one or two liners would be 
removed and the strap replaced. The key driven in would then 
draw the parts closer by the thickness of the liners removed. 

“Drawing (to be Ortho graphic, not Isometric ).—On a sheet 
14"xll" make in ink a working drawing of the front elevation of 
the rod end assembled, viewed in the direction of the arrow. Put 
paper with long dimension horizontal. Put' center of bore of 
brasses 4" from left edge of paper and 5" from top edge. Ho 
sketch, no legend, no dimensions.” 

Problems XIV. 

140. An ordinary brick measures 8"x4" x24". Make an ortho¬ 
graphic drawing and an isometric projection after the manner of 
Fig. 132, Art, 125. Contrast it with the isometric draiving, Art. 
128. 

141. Make the isometric projection of the brick, 8"x4"x24", 
turning it through the angles of 45° and 35° 44'. as in Fig. 133, 
Art, 127. 

142. From Fig. 135 make a plan and front elevation of the 
strap D. 

143. From Fig. 135 make a plan and front elevation of the stub 
end A. 

144. From Fig. 135 make a plan and front elevation of the 
brass C. 


SET OF DESCRIPTIVE DRAWINGS. 


The following four drawing sheets are designed to be executed in 
the drawing room to illustrate those principles of Descriptive 
Geometry which have the most frequent application in Mechanical 
or Engineering Drawing. 

The paper used should be about 28" X 22", the drawing-board of 
the same size, and the blade of the T-square 30". 

To lay out the sheets find the center, approximately, draw center 
lines, and draw three concentric rectangles, measuring 24" x 18", 
22"xl6", and 21"xl5". The outer rectangle is the cutting line 
to which the sheets are to be trimmed. The second one is to be inked 
for the border line. The inner one is described in pencil only as a 
“ working line,” or line outside of which no part of the actual 
figures should extend. The center lines and other fine lines, in¬ 
cluding dimensions, may extend beyond the working line. In the 
lower right corner reserve a rectangle 6" X 3", touching the working 
lines, for the legend of the drawing. 

In making the drawings three widths of line are used. 

The actual lines of the figures must be “ standard lines ” or lines 
not quite one-hundredth of an inch thick. The thin metal erasing 
shield may be used as a gauge for setting the right-line pen, by so 
adjusting the pen that the shield will slowly slip from between the 
nibs, when inserted and allowed to hang vertically. Visible edges 
are full lines. Hidden edges are broken lines; the dashes -J" long 
and spaceslong. 

The extra-fine lines are described with the pen adjusted to as fine 

a line as it will carry continuously. The axes of projection are 

fine full lines. The dimension lines are long dashes, J" to 1" long, 

with -J" spaces. The center lines are long dashes with fine dots 

between the dashes, or are dash-dot lines. The construction lines 

are long dashes with two dots between, or are dash-dot-dot lines. 

When auxiliary cutting planes are used, one only, together with its 

corresponding projection lines, should be inked in this manner. 

* 

11 


150 


Engineering Descriptive Geometry 


The extra-heavy lines are about two-hundredths of an inch thick, 
and are for two purposes: for shade lines , if used; and for paths of 
sections, or lines showing where sections have been taken, as np, 
Fig. 32. These paths of sections should be formed of dashes 
about f" long. 

SHEET I: PRISMS AND PYRAMIDS. 

Lay out the sheet and from the center of the sheet plot three ori¬ 
gins: The first origin 5f" to the left and 44" above the center of the 
sheet; the second 8" to the right and l-f" above the center; and the 
third 4" to the left and 4" below the center. Pass vertical and 
horizontal lines through these points to act as axes of projection. 

First Origin: Pentagonal Prism and Inclined Plane. 

Describe a pentagonal prism, the axis extending from P (2", 
If", 1") to P' (2", If", 2-J"). The top base is a regular pentagon 
inscribed in a circle of 1J" radius, one corner of the pentagon 
being at A (2", f", f"). Draw three views of the prism. Draw the 
traces of a plane P, perpendicular to V? its trace on V passing 
through the point c (0", 0", 2-J") and making an angle of 60° with 
the axis of Z. Draw on the side elevation the line of intersection 
of the prism and the plane P. Show the true shape of the polygonal 
line of intersection on an auxiliary plane HJ? perpendicular to V. 
its traces on V passing through the point (0", 0", 4J"). On \] 
show only the section cut by the plane. Draw the development 
of the surface of the prism, with the line of intersection described 
on it. Draw the left edge of the development [representing 
A (2", f", f"), A / (2", f", 24") ] as a vertical line 4" to the right of 
the axis of Y, and use the top working edge of the sheet as the top 
line of the development. 

Second Origin: Octagonal Prism and Triangular Prism. 

Describe an octagonal prism, the axis extending from P (2f", 
If" 4") to P f (2f", If", 4f"). The octagonal base is circumscribed 
about a circle of 2f" diameter, one flat side being parallel to the 


Set of Descriptive Drawings 


151 


axis of X. Describe a triangular prism, its axis extending from 
Q (0"52, If", If") to Q' (3f 98, If", 3J"), intersecting PP' at its 
middle point and making an angle of 60° with it. The triangular 
base is in a plane perpendicular to QQ' and is circumscribed about 
a circle of 1" diameter. One corner is at J (1", If", Of 38). Draw 
the H, V? an( i S projections of the prisms and a complete pro¬ 
jection on a plane UJ, taken perpendicular to QQ', and whose trace 
on V passes through the point (6", 0", 0"). Draw the triangular 
prism as if piercing the octagonal prism. 

Third Origin: Hexagonal Pyramid and Square Prism. 

Describe an hexagonal pyramid, vertex at P ( If", 2", f"), center 
of base at P' (If", 2", 3"). The hexagonal base is in a plane parallel 
to H and is circumscribed about a circle 2-f" in diameter, one 
corner being at A (If", Of 50, 3"). Projecting from the sides of the 
pyramid are two portions of a square prism, whose axis is Q (f", 
2", 2f"), Q' (3J", 2", 2f"). The square base is in a plane parallel 
to § and measures 1" on each edge, and its edges are parallel to the 
axes of Y and Z. Letter the edges GG', HH', etc., the point G 
being (f", If", If"), H (f", 2f", If"), etc. Draw the object as if 
cut from one solid piece of material, the prism not piercing the 
pyramid. 

The views required are plan, front elevation, and side elevation, 
and also an auxiliary projection on a plane UJ? perpendicular to J-J. 
The trace of UJ makes an angle of 120° with the axis of X at 
the point X (2§", 0", 0"). 

Draw also the developments of the surfaces. Place the vertex of 
the developed pyramid at a point f" to the right and 3f" above the 
origin, and the point A V to the right and Of 56 above the origin. 
Mark the line of intersection with the prism on this development. 

Between the side elevation and the legend space, draw the de¬ 
velopment of the square prism, placing the long edges, GG', HH', 
etc., in a vertical position. Describe the line of intersection on the 
development. Let the edge which has been opened out be GG', and 
let the middle portion of the prism, which does not in reality exist, 
be drawn with construction lines. 


152 


Engineering Descriptive Geometry 


General Directions for Completing the Sheet. 

In inking the sheet show one line of projection for the determi¬ 
nation of one point on each line of intersection. Shade the figure, 
except the developments. 

In the legend space make the following legend: 


SHEET I. 

DESCRIPTIVE GEOMETRY. 
PRISMS AND PYRAMIDS. 

Name (signature). Class. 

Date. 


(Block letters 15/32" high.) 

(All caps 3/16" high.) 

(All caps 9/32" high.) 

(Caps 1/8" high, lower case 1/12" high.) 

(Caps 1/8" high, lower case 1/12" high.) 


SHEET II: CYLINDERS, ETC. 

Lay out cutting, border, and working lines, and legend space as 
before. 

Plot four points of origin as follows: First origin, 6" to the left 
and 4" above the center of the sheet; second origin, 4f" to the right 
and 4J" above the center; third origin, 6J" to the left and 3-J" 
below the center ; fourth origin, 6f" to the right and 4f" below the 
center. 


First Origin: Intersecting Right Cylinders. 

Draw the three views of two intersecting right cylinders. The 
axis of one is P (2-J", 2", -J")> P' (2-J", 2", 3-J"), and its diameter is 
3". The axis of the other is Q (¥, If", 2"), Q' (4-J", If", 2"), and 
its diameter 2§". Determine the line of intersection in V by planes 
parallel to V at distances of f", 1", If", etc. 

Second Origin: Inclined Cylinder and Inclined Plane. 

Draw three views of an inclined circular cylinder, cut by a plane. 
The axis of the cylinder is P (3.73", If", -J")' P' (2", If", 34"). 
The base is a circle, diameter 2f", in a plane parallel to ff. The 
plane cutting the cylinder is perpendicular to V, and its trace in 
V passes through the middle point of PP\ and inclines up to the 


Set of Descriptive Drawings 


153 


left at an angle of 30° with OX. Plot the intersection in ff, V, 
and ^ and find the true shape of the ellipse by an auxiliary plane 
of projection perpendicular to V through the point (3", 0", 4"). 

Third Origin: Right Circular Cone and Inclined Plane. 

Draw a right circular cone, vertex at P (2", If", center of 
base at P' (2", If", 4"), diameter of base 3". The cone is inter¬ 
sected by a plane perpendicular to §, having its trace in § parallel 
to the extreme right element of the cone and through the point 
(0", 2J", 4"). Draw the line of intersection in plan and front 
elevation, and show the true shape of the curve by projection on an 
auxiliary plane UJ perpendicular to §, its trace passing through 
the point (0", 2J", 0"). 

Fourth Origin: Ogival Point, Vertical Plane and Inclined Plane. 

Let § lie to the right of ff and make no use of V- The problem 
is to draw two views of a 3^" ogival shell, intersected by two planes. 
The ogival point is generated by revolving 60° of arc of 3J" radius 
about an axis perpendicular to H at the point (2", If", 0"). The 
initial position of the generating arc is as follows: The center is 
at I) (0", 3-J", 3-J"), one extremity is at B (0", 0", 3J"), and one is at 
P (2", If", 0.46"). The cylindrical body of the shell extends from 
the ogival point to the right in the side elevation, a distance of f". 
Two planes, T and R, intersect the shell. T is parallel to and 1-J" 
from §. R is perpendicular to 3, and its 3 trace passes through 
the origin, and makes angles of 45° with the axis of Y and the 
axis of Z. Draw: The traces of T and R; the side elevation; the 
line of intersection of T with the shell; and, on the plan, the line 
of intersection of R with the shell. 

General Directions for Completing the Sheet. 

In inking the sheet show one cutting plane for the determination 
of each line of intersection, and show clearly how one point is de¬ 
termined in each view of each figure. Shade the figure except the 
developments. 


154 


Engineering Descriptive Geometry 


In the legend space make the following legend: 


SHEET II. 

DESCRIPTIVE GEOMETRY. 
INTERSECTIONS OF CYLINDERS,, ETC. 
Name (signature). Class. 

Date. 


(Block letters 15/32" high.) 

(All caps 3/16" high.) 

(All caps 9/32" high.) 

(Caps 1/8" high, lower case 1/12" high.) 

(Caps 1/8" high, lower case 1/12" high.) 


SHEET III: SURFACES OF REVOLUTION. 

Lay out center lines, cutting, border and working lines, and 
legend space as before. 

Plot five points of origin as follows: First origin, 6J" to the 
left and 3f" above the center of the sheet; second origin, If" to 
the right and 6" above the center; third origin, 8J" to the right and 
5f" above the center; fourth origin, 5f" to the left and 4f" below 
the center; fifth origin, 7}" to the right of the center of the sheet 
on the horizontal center line. 

First Origin: Sphere and Cylinder. 

Draw a sphere pierced by a right circular cylinder. The center 
of the sphere is at (2", 2", 2"), its diameter 3f". The axis of the 
cylinder is P (2", If", f"), P' (2", If", 3f"). Draw the sphere and 
cylinder in J-J, V and §, and determine the line of intersection by 
passing planes parallel to V at distances of -f", f", If" and If". 

Second Origin: Forked End of Connecting-Rod. 

The forked end of a connecting rod has the shape of a surface of 
revolution, faced off at the sides to a width of If", as shown in Fig. 
136. The centers a, b, and c are points (2", 1", 0"), (2", 0", f"), 
and (2", 0", 1"). The arc which has d as a center is tangent at its 
ends to the adjacent arc and to the side of the 1" cylinder. 

Determine the continuation of the line of intersection of the 
plane and surface at w, by passing planes parallel to ff at distances 
from H of 2J", 2§", 24", 2f" and 2f". Draw no side view. 








Set of Descriptive Drawings 


155 


Third Origin: Stub End of Connecting-Rod. 


The stub end of a connecting-rod is a surface of revolution faced 
off at the sides to a width of If", and pierced by bore-holes parallel 
to its axis as shown in Fig. 137. Centers are at a (If", 1", 0"), 
b (3", 1" 0"), c (f", 1", 0 "),d (3§", 0", in, and e (§", 0", If"). 
Determine the continuation of the line of intersection at w by pass¬ 
ing planes parallel to ff at distances from ff of 1 -fg", If", l T 3 g-", 



If", and l-gr". Draw also the side view and determine the ap¬ 
pearance of the edge marked u, where the large part of the bore-hole 
intersects the surface of revolution, bv means of the same system of 
planes with two additional planes. 


Fourth Origin: Right Circular Cylinder and Cone. 

A right circular cone is pierced by a right circular cylinder, the 
axes intersecting at right angles, as in Fig. 62, Art. 72. The axis 

































































156 


Engineering Descriptive Geometry 


of the cone is P (2f', 2J", J"), P f (2£", 2-J", 2f"). The base, in a 
plane parallel to is a circle of 3f" diameter. The axis of the 
cylinder is Q (£", 2J", If"), (T (4", 2-f", If"), and its diameter is 
l-i" 

Draw three views of the figures, determining the line of inter¬ 
section by planes parallel to fj. It is best not to pass these planes at 
equal intervals, but through points at equal angles on the base of 
the cylinder. Divide the base of the cylinder in § into arcs of 30°, 
and in numbering the points let that corresponding to F, in Fig. 62, 
be numbered 0 and let II be numbered 6. Insert intermediate 
points from 1 to 5 on both sides, so that the horizontal planes used 
for the determination of the curve of intersection are seven in 
number, the lowest passing through the point 0, the second through 
the two points 1, the third through the two points 2, etc. Determine 
the curve of intersection by these planes. 

Draw the development of the surface of the cylinder, cutting the 
surface on the element 00' (or FF' in Fig. 62). Place this line of 
the development vertically on the sheet, the point 0 being 1" to the 
left and 7-J" below the center of the sheet, and O' being 1" to the 
left and 3f" below the center of the sheet. 

Draw the development of the surface of the cone Note that the 
radius of the base, the altitude, and the slant height are in the 
ratio of 3:4:5. To get equally spaced elements on the surface of 
the cone, divide the arc corresponding to BO in J-J, Fig. 62, into 
five equal spaces. Number the point B 0 and C 5, and the inter¬ 
mediate points in series. Since the cone is symmetrical about two 
axes at right angles, one quadrant may represent all four quadrants. 
Put the vertex of the developed surface 5" to the left of the center 
of the sheet and 1" below it, and consider it cut on the line P 0 or 
PB. Locate the point 0 5" to the left of the center of the sheet and 
4-J" below it. Divide the development into four quadrants and then 
divide each quadrant into five parts, numbering the 21 points 
0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0. 

Fifth Origin: Cone and Double Ogival Point. 

In this figure a right circular cone pierces a double ogival point. 
The cone has a vertical axis, PP', the vertex P being at (3", 1J", 


Set of Descriptive Drawings 


157 


i"), and P', the center of the base, at (3", 1£", 3J"). The base is a 
circle of 2-J" diameter lying in a horizontal plane. 

The ogival point has an axis of revolution, Q (§", 1^", 2"), 
Q' (5f", 14", 2"), 5J" long. The generating line is an arc of 4" 
radius of which QQ' is the chord, and in its initial position the arc 
has its center at (3", 14", 5702). Draw three views of the cone 
piercing the double ogival surface, and determine the line of inter¬ 
section by means of three auxiliary cutting spheres, centered at p, 
the intersection of PP' and QQ'. Use diameters of 2-J", 2J", and 
2 t V'. This curve appears on the U. S. Navy standard 3" valve. 


General Directions for Completing the Sheet. 

In inking the sheet show one cutting plane or sphere for the 
determination of each line of intersection, and show clearly how 
one point is determined in each view of each figure. Shade the 
figures, except the developments. 

In the legend space record the following legend: 


SHEET III. 

DESCRIPTIVE GEOMETRY. 
INTERSECTIONS OF SURFACES OF 
REVOLUTION. 

Name (signature). Class. 

Date. 


(Block letters 15/32" high.) 

(All caps 3/16" high.) 

(All caps 9/32" high.) 

(Caps 1/8" high, lower case 1/12" high.) 

(Caps 1/8" high, lower case 1/12" high.) 


SHEET IV: CONES, ANCHOR RING AND HELICOIDAL 

SURFACES. 

Lay out center lines, cutting, border, working lines, and legend 
space as before. 

From the center of the sheet plot origins as follows: First 
origin, 34" to the left of the center and 3y£" above the center; 
second origin, 5-J" to the right of the center and 3" above the center; 
third origin, 104" to the right of the center and 4J" above the center; 
fifth origin, 3" to the right of the center and 6" below the center. 


158 


Engineering Descriptive Geometry 


First Origin: Intersecting Inclined Cones. 

Draw two intersecting inclined cones. The first cone has its 
vertex at P (1", If", f"), and the center of its base at P' (2ff", 
If", 4|"). The base is a circle of 3f" diameter, lying in a plane 
parallel to ff. The second cone has its vertex at Q (5", If", 2f"), 
and the center of its base at Q' (f", If", 3f"). The base is a circle 
of 3" diameter lying in a plane parallel to §. Draw plan, front 
elevation, side elevation, and an auxiliary projection on a plane U> 
perpendicular to the line PQ, the trace of U on V, passing through 
the point M (7f ", 0", 0"). Determine the line of intersection of the 
cones by auxiliary cutting planes containing the line PQ, and treat 
the problem on the supposition that the cone PP' pierces the cone 

QQT- 


Second Origin: Helieoidal Surface for Screw Propeller. 

A right vertical cylinder, If" in diameter, has for its axis 
P (2f", 2f", f"), P' (2f", 2f", 3f"). Projecting from the cylinder 
is a line A (3J", 2f", f"), B (4f", 2f", f"). This line, moving 
uniformly along the cylinder, and about it clockwise, describes one 
complete turn of a helieoidal surface of 3" pitch. Draw plan and 
front elevation of the figure. This helicoid is intersected by an 
elliptical cylinder of which the generating line is perpendicular to 
H and the directrix is an ellipse lying in fj, having its major axis 
C (2", 2f", 0"), D (f", 2f", 0"), and minor axis E (If", 3", 0"), 
F (If", 2", 0"). Find the intersection of the two surfaces. Ink in 
full lines only the circular cylinder and the intersection. This 
portion of a helieoidal surface is similar to that which is used for 
the acting surface of the ordinary marine screw propeller, of 3 or 
4 blades. 


Third Origin: Worm Thread Surface. 

A worm shaft is a right cylinder, If" in diameter, its axis being 
P (If", If", J"), P' (If", If", 8f"). A triple right-hand worm 
thread, of the same profile as in Fig. 70, projects from the cylinder 
along the middle 6" of its length. The pitch of the thread is 4f", 


Set of Descriptive Drawings 


159 


so that each thread has more than a complete turn. The outside 
diameter of the worm is 3". Make a complete drawing of the plan 
and front elevation, as in Fig. 70, letting the worm thread begin at 
any point on the circumference. 

Fourth Origin: Anchor Ring and Planes. 

An anchor ring, R, is formed by revolving a circle of 1£" diameter, 
lying in a plane parallel to V and with its center at A (1-J", 2§", 
§"), about an axis perpendicular to ff and piercing ff at the point 
B (2§", 2§", 0"). Draw plan, front elevation, right side elevation 
(to the right of IH{), and left side elevation (to the left of fi) on a 
plane S', 4J" from S- A plane P, parallel to § at f" from S, cuts 
the ring. Draw the trace of P on H, and the intersection of P and 
the ring on 5- A second plane P', parallel to § at 1-J" distance, 
cuts the ring. Draw the trace P'H and the intersection P'R on S- 
A third plane Q is parallel to V at If" distance from V* Draw the 
trace QTI and the intersection QR on V- A fourth plane, Q', is 
parallel to V at 2" distance. Draw the trace Q'H and the inter¬ 
section Q'R on V* An inclined plane T is perpendicular to § and 
S', its trace on S' passing through the point C (4f", 2f", £"), and 
inclining down to the right at such an angle as to be tangent to the 
projection on S' of the generating circle when its center is at 
D (2f", 1-J", $"). Draw T the trace of T on S', and the intersection 
TR on H. Find the true shape of TR by means of an auxiliary 
plane of projection UJ perpendicular to S', cutting S' in a trace 
parallel to TS' through the point on S' whose coordinates are 
E (4f", 0", li"). 


General Directions for Completing the Sheet. 

Ink the sheet uniform with the preceding sheets, and in the 
legend space record the following legend: 

SHEET IV. (Block letters 15/32" high.) 

DESCRIPTIVE GEOMETRY. (All caps 3/16" high.) 

CONES, ANCHOR RING AND HELICOIDS. (All caps 9/32" high.) 

Name (signature). ClaSS. (Caps 1/8" high, lower case 1/12" high.) 

Date. (Caps 1/8" high, lower case 1/12" high.) 








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